Finding the Inverse of a 2 × 2 Matrix Find the inverse (if it exists) of each matrix. a. [latex] A=\left[\begin{array}{ll}5 & 2 \\4 & 3\end{array}\right][/latex] b. [latex]B=\left[\begin{array}{ll}4 & 6 \\2 & 3\end{array}\right][/latex]

a. For the matrix [latex]A, a=5, b=2, c=4, \text { and } d=3[/latex]. Here [latex]a d-b c=(5)(3)-(2)(4)=15-8=7 \neq 0;[/latex] so A is invertible and [latex]A^{-1}=\frac{1}{7}\left[\begin{array}{rr}3 & -2 \\-4 & 5\end{array}\right] \quad \text { Substitute for } a, b, c \text {, and } d \text { in } \frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\-c & a\end{array}\right] \text {. }[/latex] [latex]=\left[\begin{array}{rr}\frac{3}{7} & -\frac{2}{7} \\-\frac{4}{7} & \frac{5}{7}\end{array}\right][/latex] Scalar multiplication You should verify that [latex]A A^{-1}=I.[/latex] b. For the matrix [latex] B, a=4, b=6, c=2, \text { and } d=3 \text {; so } a d-b c=(4)(3)-(6)(2) =12-12=0.[/latex] Therefore, the matrix B does not have an inverse.

Question 10.3.5: Finding the Inverse of a 2 × 2 Matrix Find the inverse (if i...

Precalculus A Right Triangle Approach

Finding the Inverse of a 2 × 2 Matrix

Find the inverse (if it exists) of each matrix.

a. $A=\left[\begin{array}{ll}5 & 2 \\4 & 3\end{array}\right]$ b. $B=\left[\begin{array}{ll}4 & 6 \\2 & 3\end{array}\right]$

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a. For the matrix $A, a=5, b=2, c=4, \text { and } d=3$ .

Here $a d-b c=(5)(3)-(2)(4)=15-8=7 \neq 0;$ so A is invertible and

$A^{-1}=\frac{1}{7}\left[\begin{array}{rr}3 & -2 \\-4 & 5\end{array}\right] \quad \text { Substitute for } a, b, c \text {, and } d \text { in } \frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\-c & a\end{array}\right] \text {. }$

$=\left[\begin{array}{rr}\frac{3}{7} & -\frac{2}{7} \\-\frac{4}{7} & \frac{5}{7}\end{array}\right]$ Scalar multiplication

You should verify that $A A^{-1}=I.$

b. For the matrix $B, a=4, b=6, c=2, \text { and } d=3 \text {; so } a d-b c=(4)(3)-(6)(2) =12-12=0.$ Therefore, the matrix B does not have an inverse.