Determinants & Inverses (DP IB Maths: AI HL)

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Naomi C

Last updated

22 January 2024

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Determinants

What is a determinant?

The determinant is a numerical value (positive or negative) calculated from the elements in a matrix and is used to find the inverse of a matrix
You can only find the determinant of a square matrix
The method for finding the determinant of a $2 \times 2$ matrix is given in your formula booklet:

$A = (\begin{matrix} a & b \\ c & d \end{matrix}) \Rightarrow \det A = |A| = a d - b c$

You only need to be able to find the determinant of a $2 \times 2$ matrix by hand

For larger $n \times n$ matrices you are expected to use your GDC

The determinant of an identity matrix is $\det (I) = 1$
The determinant of a zero matrix is $\det (O) = 0$
When finding the determinant of a multiple of a matrix or the product of two matrices:

$\det (k A) = k^{2} \det (A)$ (for a $2 \times 2$ matrix)
$\det (A B) = \det (A) \times \det (B)$

Worked example

Consider the matrix $A = (\begin{matrix} 3 & - 6 \\ p & 7 \end{matrix})$ , where $p \in ℝ$ is a constant.

Given that

\det A = - 3

, find the value of

p

1-7-3-ib-ai-hl-determinants--inverses-we-1a

Find the determinant of

4 A

1-7-3-ib-ai-hl-determinants--inverses-we-1b

Inverse Matrices

How do I find the inverse of a matrix?

The determinant can be used to find out if a matrix is invertible or not:
- If $\det A \neq 0$ , then $A$ is invertible
- If $\det A = 0$ , then $A$ is singular and does not have an inverse
The method for finding the inverse of a $2 \times 2$ matrix is given in your formula booklet:

$A = (\begin{matrix} a & b \\ c & d \end{matrix}) \Rightarrow A^{- 1} = \frac{1}{\det A} (\begin{matrix} d & - b \\ - c & a \end{matrix}), a d \neq b c$

You only need to be able to find the inverse of a $2 \times 2$ matrix by hand

For larger $n \times n$ matrices you are expected to use your GDC

The inverse of a square matrix $A$ is the matrix $A^{- 1}$ such that the product of these matrices is an identity matrix, $A A^{- 1} = A^{- 1} A = I$
- As a result of this property:
  - $A B = C \Rightarrow B = A^{- 1} C$ (pre-multiplying by $A^{- 1}$ )
  - $B A = C \Rightarrow B = C A^{- 1}$ (post-multiplying by $A^{- 1}$ )

Worked example

Consider the matrices $P = (\begin{matrix} 4 & - 2 \\ 8 & 2 \end{matrix})$ , $Q = (\begin{matrix} k & 6 \\ - 5 & 3 \end{matrix})$ and $R = (\begin{matrix} 18 & 18 \\ 6 & 54 \end{matrix})$ , where $k$ is a constant.

Find

P^{- 1}

1-7-3-ib-ai-hl-determinants--inverses-we-2a

Given that

P Q = R

find the value of

k

1-7-3-ib-ai-hl-determinants--inverses-we-2b

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