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IBMaths: AI HLDPRevision Notes1. Number & Algebra1.7 Matrices1.7.2 Operations with Matrices

Operations with Matrices (DP IB Maths: AI HL)

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Naomi C

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Naomi C

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Matrix Addition & Subtraction

Just as with ordinary numbers, matrices can be added together and subtracted from one another, provided that they meet certain conditions.

How is addition and subtraction performed with matrices?

  • Two matrices of the same order can be added or subtracted
  • Only corresponding elements of the two matrices are added or subtracted
    • bold italic A plus-or-minus bold italic B equals left parenthesis a subscript i j end subscript right parenthesis plus-or-minus left parenthesis b subscript i j end subscript right parenthesis equals left parenthesis a subscript i j end subscript plus-or-minus b subscript i j end subscript right parenthesis
  • The resultant matrix is of the same order as the original matrices being added or subtracted

What are the properties of matrix addition and subtraction?

  • bold italic A plus bold italic B equals bold italic B plus bold italic A(commutative)
  • 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 (associative)
  • bold italic A plus bold italic O equals bold italic A
  • bold italic O minus bold italic A equals negative bold italic A
  • bold italic A minus bold italic B equals bold italic A plus left parenthesis negative bold italic B right parenthesis

Examiner Tip

  • Make sure that you know how to add and subtract matrices on your GDC for speed or for checking work in an exam!

Worked example

Consider the matrices bold italic A equals open parentheses table row cell negative 4 end cell 2 row 7 3 row 1 cell negative 5 end cell end table close parentheses, bold italic B equals open parentheses table row 2 6 row 5 cell negative 9 end cell row cell negative 2 end cell cell negative 3 end cell end table close parentheses.

a)
Find bold italic A plus bold italic B.

1-7-2-ib-ai-hl-operations-with-matrices-we-1a-solution

b)
Find bold italic A minus bold italic B.

1-7-2-ib-ai-hl-operations-with-matrices-we-1b-solution

Matrix Multiplication

Matrices can also be multiplied either by a scalar or by another matrix.

How do I multiply 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
  • The resultant matrix is of the same order as the original matrix
  • Multiplication by a negative scalar changes the sign of each element in the matrix

How do I multiply a matrix by another matrix?

  • 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
  • If the order of the first matrix is m cross times n and the order of the second matrix is n cross times p, then the order of the resultant matrix will be m cross times p
  • 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 
      • then  bold italic B bold italic A equals open square brackets table row cell left parenthesis g a plus h d right parenthesis end cell cell left parenthesis g b plus h e right parenthesis end cell cell left parenthesis g c plus h f right parenthesis end cell row cell left parenthesis i a plus j d right parenthesis end cell cell left parenthesis i b plus j e right parenthesis end cell cell left parenthesis i c plus j f right parenthesis end cell row cell left parenthesis k a plus l d right parenthesis end cell cell left parenthesis k b plus l e right parenthesis end cell cell left parenthesis k c plus l f right parenthesis end cell end table close square brackets

How do I square an expression involving matrices?

  • If an expression involving matrices is squared then you are multiplying the expression by itself, so write it out in bracket form first, e.g. left parenthesis A plus B right parenthesis squared equals left parenthesis A plus B right parenthesis left parenthesis A plus B right parenthesis
    • remember, the regular rules of algebra do not apply here and you cannot expand these brackets, instead, add together the matrices inside the brackets and then multiply the matrices together

What are the properties of matrix multiplication?

  • bold italic A bold italic B not equal to bold italic B bold italic A (non-commutative)
  • bold italic A left parenthesis bold italic B bold italic C right parenthesis equals left parenthesis bold italic A bold italic B right parenthesis bold italic C (associative)
  • 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 (distributive)
  • left parenthesis bold italic A plus bold italic B right parenthesis bold italic C equals bold italic A bold italic C plus bold italic B bold italic C (distributive)
  • bold italic A bold italic I equals bold italic I bold italic A equals bold italic A (identity law)
  • bold italic A bold italic O equals bold italic O bold italic A equals bold italic O, where bold italic O is a zero matrix
  • Powers of square matrices: bold italic A squared equals bold italic A bold italic A comma space bold italic A cubed equals bold italic A bold italic A bold italic A etc.

Examiner Tip

  • Make sure that you are clear on the properties of matrix algebra and show each step of your calculations

Worked example

Consider the matrices bold italic A equals open square brackets table row 4 2 cell negative 5 end cell row cell negative 3 end cell 8 1 row cell negative 1 end cell cell negative 2 end cell 2 end table close square brackets and bold italic B equals open square brackets table row 5 1 row cell negative 2 end cell 5 row 9 7 end table close square brackets .

a)
Find bold italic A bold italic B.

1-7-2-ib-ai-hl-operations-with-matrices-we-2a-solution

b)
Explain why you cannot find bold italic B bold italic A.

1-7-2-ib-ai-hl-operations-with-matrices-we-2b-solution

c)
Find bold italic A squared.

rn-1-7-matrices

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    Naomi C

    Author: Naomi C

    Expertise: Maths

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.