Question

1 - 0: find the inverse of each matrix algebraically or explain why it cannot be found. show your work for full credit, and check your answers with a calculator.
(a) $a=\begin{bmatrix}-7&5\\4& - 3end{bmatrix}$
(b) $b=\begin{bmatrix}4&5\\2&3end{bmatrix}$

Explanation:

Step1: Recall the formula for 2x2 matrix inverse

For a 2x2 matrix $M=

$$\begin{bmatrix}a&b\\c&d\end{bmatrix}$$

$, the inverse $M^{-1}=\frac{1}{ad - bc}

$$\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$$

$, provided $ad - bc
eq0$.

Step2: Calculate the inverse of matrix A

For $A=

$$\begin{bmatrix}-7&5\\4&-3\end{bmatrix}$$

$, first find the determinant $ad - bc=(-7)\times(-3)-5\times4 = 21 - 20=1$.
Then $A^{-1}=\frac{1}{1}

$$\begin{bmatrix}-3&-5\\-4&-7\end{bmatrix}$$

=

$$\begin{bmatrix}-3&-5\\-4&-7\end{bmatrix}$$

$.

Step3: Calculate the inverse of matrix B

For $B=

$$\begin{bmatrix}4&5\\2&3\end{bmatrix}$$

$, find the determinant $ad - bc=4\times3 - 5\times2=12 - 10 = 2$.
Then $B^{-1}=\frac{1}{2}

$$\begin{bmatrix}3&-5\\-2&4\end{bmatrix}$$

=

$$\begin{bmatrix}\frac{3}{2}&-\frac{5}{2}\\-1&2\end{bmatrix}$$

$.

Answer:

(a) $A^{-1}=

$$\begin{bmatrix}-3&-5\\-4&-7\end{bmatrix}$$

$
(b) $B^{-1}=

$$\begin{bmatrix}\frac{3}{2}&-\frac{5}{2}\\-1&2\end{bmatrix}$$

$