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14. Matrices and Networks
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Middle Years

14.04 Inverse of a matrix

Lesson
Worksheet
Practice
Worksheet
Determinants of 2 x 2 matrices
1

Evaluate the following:

a
\begin{vmatrix} 1 & 7 \\ 9 & 5\end{vmatrix}
b
\begin{vmatrix} -4 & -6 \\ 3 & 1\end{vmatrix}
c
\begin{vmatrix} 11 & 13 \\ 12 & 14\end{vmatrix}
d
\begin{vmatrix} \dfrac{1}{3} & -15 \\ \\ \dfrac{4}{5} & -6\end{vmatrix}
e
\begin{vmatrix} 4.6 & -5.9 \\ 9.5 & -1.5\end{vmatrix}
f
\begin{vmatrix} \dfrac{\pi}{2} & 3 \\ \\-\dfrac{1}{3} & -4\end{vmatrix}
2

Form an expression for the determinant of the matrix \begin{bmatrix} -5n & 2n^2 \\ -7 & 9n\end{bmatrix}.

3

If B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22}\end{bmatrix} is the inverse of A=\begin{bmatrix} -8 & -9\pi \\ \dfrac{1}{3} & \dfrac{1}{4}\end{bmatrix}, find:

a

\text{det}(A)

b

b_{11}

c

b_{12}

d

b_{21}

e

b_{22}

4

Consider the equation \begin{vmatrix} 9 & 6 \\ 3 & n\end{vmatrix}= 18. Solve for n.

Inverses of 2 x 2 matrices
5

For each matrix A:

i
Find the determinant of A.
ii
Find A^{-1}.
a
A=\begin{bmatrix} 3 & 9 \\ 2 & 7\end{bmatrix}
b
A=\begin{bmatrix} -4 & 7 \\ 2 & -3\end{bmatrix}
c
A=\begin{bmatrix} 11 & 17 \\ 13 & 22\end{bmatrix}
d
A=\begin{bmatrix} 9 & 2 \\ 3 & 7\end{bmatrix}
e
A=\begin{bmatrix} 2 & 1 \\ -3 & -9\end{bmatrix}
f
A=\begin{bmatrix} 25 & 26 \\ 21 & 22\end{bmatrix}
g
A=\begin{bmatrix} -\dfrac{1}{3} & \dfrac{1}{3} \\\\ 9 & -6\end{bmatrix}
h
A=\begin{bmatrix} -9.7 & -3.2 \\ -7.2 & -4.7 \end{bmatrix}
i
A = \begin{bmatrix} -6 & -4 \\ \\\dfrac {1}{2} & \dfrac{\pi}{3}\end{bmatrix}
6

Determine the inverse of the following matrices:

a
B=\begin{bmatrix} -7 & -1 \\ 9 & 3\end{bmatrix}
b
V=\begin{bmatrix} 16 & 21 \\ 12 & 13\end{bmatrix}
c
X=\begin{bmatrix} -8 & -\dfrac{1}{2} \\\\ 15 & \dfrac{2}{5}\end{bmatrix}
d
P=\begin{bmatrix} 3 & 4 \\ 1 & 9\end{bmatrix}
e
C=\begin{bmatrix} -7.6 & 9.5 \\ -5.2 & -2.9\end{bmatrix}
7
a

State the condition for a matrix to have an inverse.

b

Hence determine if the matrix \begin{bmatrix} 4 & 2 \\ -5 & 6\end{bmatrix} has an inverse.

8

The following matrices are inverses:

X= \begin{bmatrix} 7 & 4 \\ 5 & 3\end{bmatrix} \text{ and } Y=\begin{bmatrix} 3 & -4 \\ -5 & 7\end{bmatrix}

a

Calculate XY.

b

Calculate YX.

9

For each pair of matrices:

i

Find AB.

ii

Hence state whether A and B are inverses of each other.

a
A = \begin{bmatrix} 7 & 2 \\ 3 & 1\end{bmatrix} \text{ and } B=\begin{bmatrix} 1 & -2 \\ -3 & 7\end{bmatrix}
b
A = \begin{bmatrix} 7 & 3 \\ 5 & 2\end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & -3 \\ -5 & 7\end{bmatrix}
10

Let A = \begin{bmatrix} \dfrac {9}{2} & -\dfrac {5}{2} \\ \\9 & -\dfrac {9}{4}\end{bmatrix} and B=\begin{bmatrix} -2 & -1 \\ -\dfrac {7}{4} & -\dfrac {1}{3} \end{bmatrix}.

a

Find A^{ - 1 }.

b

Find X, if AX=B.

11

Consider A = \begin{bmatrix} -2 & 5 \\ -1 & 2\end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & -5 \\ 1 & -2\end{bmatrix}.

a

Find AB.

b

Find A^{-1}.

12

Given the matrix A and its inverse A^{-1}, find n:

a

A = \begin{bmatrix} 7 & 2 \\ 8 & 1\end{bmatrix} and A^{-1} =\begin{bmatrix} -\dfrac {1}{9} & \dfrac{2} {9} \\\\ \dfrac {8}{9} & n \end{bmatrix}

b

A = \begin{bmatrix} -1 & 2\\ 3 & -9\end{bmatrix} and A^{-1}=\begin{bmatrix} -3 & n \\ -1 & -\dfrac{1}{3}\end{bmatrix}

c

A = \begin{bmatrix} 5 & n \\ -2 & -4\end{bmatrix} and A^{-1} =\begin{bmatrix} \dfrac {2}{3} & \dfrac{7}{6} \\\\ -\dfrac {1}{3} & -\dfrac{5}{6}\end{bmatrix}

d

A = \begin{bmatrix} 7 & 5\\ 3 & n\end{bmatrix} and A^{-1}=\begin{bmatrix} \dfrac{3}{16} & -\dfrac{5}{48} \\\\ -\dfrac{1}{16} & \dfrac{7}{48}\end{bmatrix}

Solve matrix equations
13

Let A,B and C be matrices. Using matrix algebra, solve for matrix B in the following equations:

a
A B = C
b
B A = C
c
A B + C = 0
d
B A - C = 0
e
A B + C B = D
14

Find the matrix X for the given matrices and equations:

a

P = \begin{bmatrix} 13 & 3 \\ 7 & 11\end{bmatrix}, PX = P

b

N = \begin{bmatrix} 2 & -9 \\ -1 & -4\end{bmatrix} and I=\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}, X N = I

c

M = \begin{bmatrix} 1 & -5 \\ -5 & 6\end{bmatrix} and N=\begin{bmatrix} 7 & 8 \\ 9 & 1\end{bmatrix}, XM = N

d

M = \begin{bmatrix} 10 & -1 \\ -9 & 4\end{bmatrix} and N=\begin{bmatrix} 7 & -6 \\ 5 & -9\end{bmatrix}, M X = N

e

M = \begin{bmatrix} -6 & 10 \\ -6 & -4\end{bmatrix} and P=\begin{bmatrix} -6 & -10 \\ 4 & 1\end{bmatrix}, X P = M

f

M = \begin{bmatrix} 16 & -10 \\ 16 & 2\end{bmatrix} and N=\begin{bmatrix} -6 & -12 \\ -18 & -3\end{bmatrix}, M X = N

g

A = \begin{bmatrix} 10 & 3 \\8& 4 \end{bmatrix} and P=\begin{bmatrix} 5 \\ 6 \end{bmatrix}, A X = P

h

B = \begin{bmatrix} -2 & -3 \\ 3 & -10 \end{bmatrix} and Q=\begin{bmatrix} 5 \\ 6 \end{bmatrix}, B X = Q

i

A = \begin{bmatrix} -8 & 5 & 7 \\ 5 & 6 & -8 \\ 7 & 2 & 7 \end{bmatrix} and B=\begin{bmatrix} -10 \\ 1 \\ -8 \end{bmatrix}, A X = B

j

M = \begin{bmatrix} -2 & -8 \\ -1 & -6.6\end{bmatrix} and N=\begin{bmatrix} 5.4 & 1.6 \\ 4 & -3.1\end{bmatrix}, M X = N

k

M = \begin{bmatrix} -3 & -5 \\ 3 & 0\end{bmatrix}, N=\begin{bmatrix} 5 & 2 \\ 0 & -2 \end{bmatrix} and P=\begin{bmatrix} 6 & 10 \\ 10 & -9 \end{bmatrix}, MNX = P

l

M = \begin{bmatrix} 3 & 1 \\ -4 & 5\end{bmatrix}, N=\begin{bmatrix} 2 & -3 \\ 3 & -5 \end{bmatrix} and P=\begin{bmatrix} -6 & 10 \\ 5 & 5 \end{bmatrix}, M^{ - 1 } N X = P

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