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IBMathsDPApplications & Interpretation (AI)HLFlashcards1. Number & AlgebraMatrices

Matrices (DP IB Applications & Interpretation (AI))

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  • What is a matrix?

    A matrix is a rectangular array of elements (numerical or algebraic) that are arranged in rows and columns.

    E.g. open parentheses table row cell table row 2 cell negative 1 end cell 7 row 5 0 6 row 9 3 cell negative 3 end cell row 1 7 11 end table end cell end table close parentheses

  • True or False?

    A 3 cross times 4 matrix has 3 columns and 4 rows.

    False.

    A 3 cross times 4 matrix does not have 3 columns and 4 rows.

    A matrix of order m cross times n has m rows and n columns, so a 3 cross times 4 matrix has 3 rows and 4 columns, e.g. open parentheses table row 1 2 3 4 row 4 3 2 1 row 3 2 4 1 end table close parentheses

  • For a matrix bold italic A, defined by bold italic A equals left parenthesis a subscript i j end subscript right parenthesis, where i equals 1 comma space 2 comma space 3 comma space... comma space m and j equals 1 comma space 2 comma space 3 comma space... comma space n, which element does a subscript 2 comma 3 end subscript refer to?

    For a matrix bold italic A, defined by bold italic A equals left parenthesis a subscript i j end subscript right parenthesis, where i equals 1 comma space 2 comma space 3 comma space... comma space m and j equals 1 comma space 2 comma space 3 comma space... comma space n, the element indicated by a subscript 2 comma 3 end subscript is the element in the second row and the third column.

  • What is a column matrix?

    A column matrix (or column vector) is a matrix with a single column, n equals 1.

    E.g. open parentheses table row cell table row 3 row cell negative 2 end cell row 5 end table end cell end table close parentheses

  • What is a row matrix?

    A row matrix is a matrix with a single row, m equals 1.

    E.g. open parentheses table row cell table row 3 0 cell negative 1 end cell cell negative 4 end cell end table end cell end table close parentheses

  • What can be said about the number or rows and columns in a square matrix?

    A square matrix is one in which the number of rows is equal to the number of columns, m equals n.

  • Under what conditions are two matrices considered to be equal?

    The conditions under which two matrices are considered to be equal, are::

    1. The matrices are of the same order, e.g. both are 3 x 2 matrices.

    2. The corresponding elements in both matrices are equal, i.e. a subscript i j end subscript equals b subscript i j end subscriptfor all elements.

  • What is a zero matrix?

    A zero matrix, bold italic O, is a matrix in which all the elements are 0, e.g. bold italic O equals open parentheses table row 0 0 row 0 0 end table close parentheses.

  • True or False?

    The matrix open parentheses table row 0 1 row 1 0 end table close parentheses is known as the identity matrix, bold italic I.

    False.

    The matrix open parentheses table row 0 1 row 1 0 end table close parentheses is not the identity matrix, bold italic I.

    The identity matrix is a square matrix in which all elements along the leading diagonal are 1 and the rest are 0, e.g. bold italic I equals open parentheses table row 1 0 row 0 1 end table close parentheses.

  • What must be the same for two matrices for it to be possible to add or subtract them?

    In order to be able to add or subtract two matrices, they must be of the same order, i.e. they must have the same number of rows and columns.

  • What is the size of the resultant matrix when two 3 x 4 matrices are added?

    A resultant matrix is of the same order as the original matrices being added or subtracted, so when two 3 x 4 matrices are added, the resultant matrix is also 3 x 4.

  • True or False?

    In the context of matrices, bold italic A plus bold italic B not equal to bold italic B plus bold italic A.

    False.

    bold italic A plus bold italic B bold equals bold italic B plus bold italic A

    E.g.

    open parentheses table row cell table row 2 7 row cell negative 1 end cell 3 end table end cell end table close parentheses plus open parentheses table row cell table row 0 cell negative 4 end cell row 1 cell negative 2 end cell end table end cell end table close parentheses equals open parentheses table row cell table row cell open parentheses 2 plus 0 close parentheses end cell cell open parentheses 7 plus open parentheses negative 4 close parentheses close parentheses end cell row cell open parentheses negative 1 plus 1 close parentheses end cell cell open parentheses 3 plus open parentheses negative 2 close parentheses close parentheses end cell end table end cell end table close parentheses equals open parentheses table row cell table row 2 3 row 0 1 end table end cell end table close parentheses open parentheses table row cell table row 0 cell negative 4 end cell row 1 cell negative 2 end cell end table end cell end table close parentheses plus open parentheses table row cell table row 2 7 row cell negative 1 end cell 3 end table end cell end table close parentheses equals open parentheses table row cell table row cell open parentheses 0 plus 2 close parentheses end cell cell open parentheses negative 4 plus 7 close parentheses end cell row cell open parentheses 1 plus open parentheses negative 1 close parentheses close parentheses end cell cell open parentheses negative 2 plus 3 close parentheses end cell end table end cell end table close parentheses equals open parentheses table row cell table row 2 3 row 0 1 end table end cell end table close parentheses

  • True or False?

    Subtracting a matrix from another matrix is the same as adding its negative.

    True.

    Subtracting a matrix from another matrix is the same as adding its negative, bold italic A minus bold italic B equals bold italic A plus left parenthesis negative bold italic B right parenthesis

    E.g.

    open parentheses table row cell table row 2 7 row cell negative 1 end cell 3 end table end cell end table close parentheses minus open parentheses table row cell table row 0 cell negative 4 end cell row 1 cell negative 2 end cell end table end cell end table close parentheses equals open parentheses table row cell table row cell open parentheses 2 minus 0 close parentheses end cell cell open parentheses 7 minus open parentheses negative 4 close parentheses close parentheses end cell row cell open parentheses negative 1 minus 1 close parentheses end cell cell open parentheses 3 minus open parentheses negative 2 close parentheses close parentheses end cell end table end cell end table close parentheses equals open parentheses table row cell table row 2 11 row cell negative 2 end cell 5 end table end cell end table close parentheses open parentheses table row cell table row 2 7 row cell negative 1 end cell 3 end table end cell end table close parentheses plus open parentheses table row cell table row 0 4 row cell negative 1 end cell 2 end table end cell end table close parentheses equals open parentheses table row cell table row cell open parentheses 2 plus 0 close parentheses end cell cell open parentheses 7 plus 4 close parentheses end cell row cell open parentheses negative 1 plus open parentheses negative 1 close parentheses close parentheses end cell cell open parentheses 3 plus 2 close parentheses end cell end table end cell end table close parentheses equals open parentheses table row cell table row 2 11 row cell negative 2 end cell 5 end table end cell end table close parentheses

  • True or False?

    In the context of adding matrices, bold italic A plus left parenthesis bold italic B plus bold italic C right parenthesis equals left parenthesis bold italic A plus bold italic B right parenthesis plus bold italic C.

    True.

    bold italic A plus left parenthesis bold italic B plus bold italic C right parenthesis equals left parenthesis bold italic A plus bold italic B right parenthesis plus bold italic C

    E.g.

    open parentheses table row 2 row 8 end table close parentheses plus open parentheses open parentheses table row cell negative 4 end cell row 6 end table close parentheses plus open parentheses table row 7 row 3 end table close parentheses close parentheses equals open parentheses table row 2 row 8 end table close parentheses plus open parentheses table row 3 row 9 end table close parentheses equals open parentheses table row 5 row 17 end table close parentheses open parentheses open parentheses table row 2 row 8 end table close parentheses plus open parentheses table row cell negative 4 end cell row 6 end table close parentheses close parentheses plus open parentheses table row 7 row 3 end table close parentheses equals open parentheses table row cell negative 2 end cell row 14 end table close parentheses plus open parentheses table row 7 row 3 end table close parentheses equals open parentheses table row 5 row 17 end table close parentheses

  • When a matrix, bold italic A, is added to the zero matrix, bold italic O, what is the resultant matrix?

    When a matrix, bold italic A, is added to the zero matrix, bold italic O, the resultant matrix is bold italic A,bold italic A plus bold italic O equals bold italic A.

    It is also the case that when a matrix, bold italic A, is subtracted from the zero matrix, bold italic O, the resultant matrix is italic minus bold italic A, bold italic O italic minus bold italic A equals negative bold italic A.

  • How do you multiply a matrix by a scalar, e.g. 3 open parentheses table row 2 6 row 7 cell negative 4 end cell row cell negative 2 end cell 4 end table close parentheses?

    When multiplying a matrix by a scalar, multiply each element in the matrix by the scalar value, k bold italic A equals left parenthesis k a subscript i j end subscript right parenthesis.

    E.g. 3 open parentheses table row 2 6 row 7 cell negative 4 end cell row cell negative 2 end cell 4 end table close parentheses equals open parentheses table row 6 18 row 21 cell negative 12 end cell row cell negative 6 end cell 12 end table close parentheses

    Remember, multiplication by a negative scalar changes the sign of each element in the matrix.

  • True or False?

    To multiply a matrix by another matrix, the number of columns in the first matrix must be equal to the number of columns in the second matrix.

    False.

    To multiply a matrix by another matrix, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

    E.g. open parentheses table row 1 2 row 3 4 row 5 6 end table close parentheses cross times open parentheses table row 7 8 9 row 1 2 3 end table close parentheses

  • When multiplying two matrices, if the order of the first matrix is m cross times n and the order of the second matrix is n cross times p, what will the order of the resultant matrix be?

    When multiplying two matrices, if the order of the first matrix is m cross times n and the order of the second matrix is n cross times p, the order of the resultant matrix will be m cross times p.

  • How do you find the product of two matrices?

    The product of two matrices is found by:

    • multiplying the corresponding elements in the row of the first matrix with the corresponding elements in the column of the second matrix,

    • and finding the sum to place in the resultant matrix.

    E.g. if bold italic A equals open square brackets table row a b c row d e f end table close square brackets, bold italic B equals open square brackets table row g h row i j row k l end table close square brackets

    then bold italic A bold italic B equals open square brackets table row cell left parenthesis a g plus b i plus c k right parenthesis end cell cell left parenthesis a h plus b j plus c l right parenthesis end cell row cell left parenthesis d g plus e i plus f k right parenthesis end cell cell left parenthesis d h plus e j plus f l right parenthesis end cell end table close square brackets .

  • True or False?

    In the context of matrices, bold italic A bold italic B not equal to bold italic B bold italic A.

    True.

    bold italic A bold italic B not equal to bold italic B bold italic A

    E.g.

    open parentheses table row cell table row 3 row cell negative 1 end cell end table end cell end table close parentheses cross times open parentheses table row 5 8 end table close parentheses equals open parentheses table row cell open parentheses 3 cross times 5 close parentheses end cell cell open parentheses 3 cross times 8 close parentheses end cell row cell open parentheses negative 1 cross times 5 close parentheses end cell cell open parentheses negative 1 cross times 8 close parentheses end cell end table close parentheses equals open parentheses table row 15 24 row cell negative 5 end cell cell negative 8 end cell end table close parentheses open parentheses table row 5 8 end table close parentheses cross times open parentheses table row cell table row 3 row cell negative 1 end cell end table end cell end table close parentheses equals open parentheses table row cell 5 cross times 3 plus 8 cross times open parentheses negative 1 close parentheses end cell end table close parentheses equals open parentheses table row 7 end table close parentheses

  • True or False?

    In the context of matrices, bold italic A left parenthesis bold italic B plus bold italic C right parenthesis equals bold italic A bold italic B plus bold italic A bold italic C.

    True

    bold italic A left parenthesis bold italic B plus bold italic C right parenthesis equals bold italic A bold italic B plus bold italic A bold italic C

    E.g. open parentheses table row cell table row cell negative 2 end cell row cell negative 9 end cell end table end cell end table close parentheses open parentheses open parentheses table row cell table row 8 cell negative 3 end cell end table end cell end table close parentheses plus open parentheses table row cell table row 1 cell negative 2 end cell end table end cell end table close parentheses close parentheses equals open parentheses table row cell table row cell negative 2 end cell row cell negative 9 end cell end table end cell end table close parentheses open parentheses table row cell table row 8 cell negative 3 end cell end table end cell end table close parentheses plus open parentheses table row cell table row cell negative 2 end cell row cell negative 9 end cell end table end cell end table close parentheses open parentheses table row cell table row 1 cell negative 2 end cell end table end cell end table close parentheses

  • What is the resultant matrix for the product of a matrix, bold italic A, and the zero matrix, bold italic O?

    The resultant matrix for the product of a matrix, bold italic A, and the zero matrix, bold italic O is the zero matrix, bold italic O, bold italic A bold italic O bold equals bold italic O bold italic A bold equals bold italic O.

  • In the context of matrices, what is the identity law?

    The identity law states that when any square matrix is multiplied by the identity matrix, the result is always the original matrix, bold italic A bold italic I equals bold italic I bold italic A equals bold italic A.

  • What is the determinant of a matrix?

    The determinant of a matrix is a single numerical value (positive or negative) that is calculated from the elements in a matrix.

    The determinant is used to find the inverse of a matrix.

  • True or False?

    The determinant can be calculated for any matrix.

    False.

    The determinant can not be calculated for any matrix.

    It can only be calculated for square matrices.

  • What is the equation for the determinant of a 2 cross times 2 matrix?

    The equation for the determinant of a 2 cross times 2 matrix is:

    bold italic A equals open parentheses table row a b row c d end table close parentheses rightwards double arrow det space bold italic A equals open vertical bar bold italic A close vertical bar equals a d minus b c

    This is given in the exam formula booklet.

  • What is the determinant of an identity matrix?

    The determinant of an identity matrix is det space left parenthesis bold italic I right parenthesis equals 1.

  • What is the determinant of a zero matrix?

    What is the determinant of a zero matrix is det space left parenthesis bold italic O right parenthesis equals 0.

  • True or False?

    In the context of matrices, det space left parenthesis bold italic A bold italic B right parenthesis equals det space left parenthesis bold italic A right parenthesis cross times det space left parenthesis bold italic B right parenthesis

    True.

    det space left parenthesis bold italic A bold italic B right parenthesis equals det space left parenthesis bold italic A right parenthesis cross times det space left parenthesis bold italic B right parenthesis

  • What must be true about the determinant if a matrix, bold italic A, is invertible.

    If a matrix, bold italic A, is invertible, then its determinant must not be equal to zero, det open parentheses bold italic A close parentheses not equal to 0

  • What is the equation for finding the inverse of a 2 cross times 2 matrix?

    The equation for finding the inverse of a 2 cross times 2 matrix is:

    bold italic A equals open parentheses table row a b row c d end table close parentheses rightwards double arrow bold italic A to the power of bold minus bold 1 end exponent equals fraction numerator 1 over denominator det space bold italic A end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses comma space a d not equal to b c

    This is given in your exam formula booklet.

  • What matrix is the result of the product of a square matrix and its inverse?

    The product of a square matrix, bold italic A, and its inverse, bold italic A to the power of bold minus bold 1 end exponent, is an identity matrix, bold italic A bold italic A to the power of negative 1 end exponent equals bold italic A to the power of negative 1 end exponent bold italic A equals bold italic I.

  • True or False?

    bold italic A bold italic B equals bold italic C rightwards double arrow bold italic B equals bold italic A to the power of negative 1 end exponent bold italic C

    True.

    bold italic A bold italic B equals bold italic C rightwards double arrow bold italic B equals bold italic A to the power of negative 1 end exponent bold italic C

    This is known as pre-multiplying by an inverse matrix to find an unknown matrix and is a result of the property bold italic A bold italic A to the power of negative 1 end exponent equals bold italic A to the power of negative 1 end exponent bold italic A equals bold italic I.

    You can also post-multiply by an inverse matrix, bold italic B bold italic A equals bold italic C rightwards double arrow bold italic B equals bold italic C bold italic A to the power of negative 1 end exponent.

  • What is the general form of a system of linear equations involving a matrix?

    The general form of a linear equation involving a matrix is bold italic A x equals b, where bold italic A is the matrix of coefficients.

  • What must be true about a matrix in a system of linear equations to have a unique solution?

    For a system of linear equations to have a unique solution, the matrix of coefficients must be invertible and therefore must be a square matrix.

  • After writing a system of linear equations in matrix form, e.g. open parentheses table row 2 cell negative 3 end cell row 5 4 end table close parentheses open parentheses table row x row y end table close parentheses equals open parentheses table row cell negative 14 end cell row 11 end table close parentheses, what is the next step in solving the system?

    After writing a system of linear equations in matrix form, the next step in solving the system is to rewrite the equation using the inverse of the matrix of coefficients.

    E.g. open parentheses table row 2 cell negative 3 end cell row 5 4 end table close parentheses open parentheses table row x row y end table close parentheses equals open parentheses table row cell negative 14 end cell row 11 end table close parentheses becomes open parentheses table row x row y end table close parentheses equals open parentheses table row cell 4 over 23 end cell cell 3 over 23 end cell row cell negative 5 over 23 end cell cell 2 over 23 end cell end table close parentheses open parentheses table row cell negative 14 end cell row 11 end table close parentheses.

  • True or False?

    You must be able use matrices to solve by hand, a system of linear equations for 2 variables.

    True.

    You must be able use matrices to solve by hand, a system of linear equations for 2 variables.

    You must be able use a mixture of matrices and technology to solve a system of linear equations for up to 3 variables,

Previous:Further Complex NumbersNext:Eigenvalues & Eigenvectors

    1. Topic 1

      Number Toolkit

    2. Topic 2

      Exponentials & Logs

    3. Topic 3

      Sequences & Series

    4. Topic 4

      Financial Applications

    5. Topic 5

      Complex Numbers

    6. Topic 6

      Further Complex Numbers

    7. Topic 7

      Matrices

    8. Topic 8

      Eigenvalues & Eigenvectors

    1. Topic 1

      Linear Functions & Graphs

    2. Topic 2

      Further Functions & Graphs

    3. Topic 3

      Modelling with Functions

    4. Topic 4

      Functions Toolkit

    5. Topic 5

      Transformations of Graphs

    6. Topic 6

      Modelling with Logarithmic, Logistic & Piecewise Functions

    1. Topic 1

      Geometry Toolkit

    2. Topic 2

      Geometry of 3D Shapes

    3. Topic 3

      Trigonometry

    4. Topic 4

      Trigonometric Identities & Equations

    5. Topic 5

      Voronoi Diagrams

    6. Topic 6

      Matrix Transformations

    7. Topic 7

      Vector Properties

    8. Topic 8

      Vector Equations of Lines

    9. Topic 9

      Modelling with Vectors

    10. Topic 10

      Graph Theory

    1. Topic 1

      Statistics Toolkit

    2. Topic 2

      Correlation & Regression

    3. Topic 3

      Non-linear Regression

    4. Topic 4

      Probability

    5. Topic 5

      Probability Distributions

    6. Topic 6

      Random Variables

    7. Topic 7

      Binomial Distribution

    8. Topic 8

      Normal Distribution

    9. Topic 9

      Combinations of Normal Distributions & Sample Mean Distributions

    10. Topic 10

      Poisson Distribution

    11. Topic 11

      Hypothesis Testing using the Chi-squared Distribution

    12. Topic 12

      Hypothesis Testing for Population Parameters

    13. Topic 13

      Transition Matrices & Markov Chains

    1. Topic 1

      Differentiation

    2. Topic 2

      Further Differentiation

    3. Topic 3

      Integration

    4. Topic 4

      Further Integration

    5. Topic 5

      Kinematics

    6. Topic 6

      Differential Equations

    7. Topic 7

      Coupled & Second Order Differential Equations