Find the inverse of each matrix. $\newline$ $4$ ) $\left[\begin{array}{cc}-5 & 3 \\ -3 & -4\end{array}\right]$

Full solution

Q. Find the inverse of each matrix.

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4

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\left[\begin{array}{cc}-5 & 3 \\ -3 & -4\end{array}\right]

Write Matrix Inverse Formula: Write down the matrix and its formula for the inverse. $\newline$ The inverse of a $2 \times 2$ matrix $\left[\begin{array}{cc} a & b \ c & d \end{array}\right]$ is given by the formula $\frac{1}{ad - bc} \times \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right]$ . $\newline$ For the matrix $\left[\begin{array}{cc} -5 & 3 \ -3 & -4 \end{array}\right]$ , we have $a = -5$ , $b = 3$ , $c = -3$ , and $d = -4$ .
Calculate Determinant: Calculate the determinant $ad - bc$ . $\newline$ Determinant = $(-5)(-4) - (3)(-3)$ $\newline$ Determinant = $20 + 9$ $\newline$ Determinant = $29$
Check Determinant: Check if the determinant is zero. $\newline$ If the determinant is zero, the matrix does not have an inverse. $\newline$ Since the determinant is $29$ , which is not zero, the matrix does have an inverse.
Write Inverse Formula: Write down the formula for the inverse using the determinant. $\newline$ Inverse = $\frac{1}{29} \times \left[\begin{array}{cc} -4 & -3 \ 3 & -5 \end{array}\right]$
Multiply by Scalar: Multiply each element of the matrix by the scalar $\frac{1}{29}$ . Inverse = $\left[\begin{array}{cc} -\frac{4}{29} & -\frac{3}{29} \ \frac{3}{29} & -\frac{5}{29} \end{array}\right]$