It can be shown that the inverse of matrix A=[ a b c d ] is A^-1=[ (d)/(a

step 1 The topic of this question is matrix inverses. Recall that given a matrix A with entries A, B, C

It can be shown that the inverse of matrix A=[ a b ...

It can be shown that the inverse of matrix $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is $$A^{-1}=\left[\begin{array}{cc}\frac{d}{a d-b c} & \frac{-b}{a d-b c} \\\frac{-c}{a d-b c} & \frac{a}{a d-b c}\end{array}\right]$$ Work Exercises in order. Use the method described here to find the inverse of $A=\left[\begin{array}{ll}4 & 2 \\ 7 & 3\end{array}\right].$

It can be shown that the inverse of matrix $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is $$A^{-1}=\left[\begin{array}{cc}\frac{d}{a d-b c} & \frac{-b}{a d-b c} \\\frac{-c}{a d-b c} & \frac{a}{a d-b c}\end{array}\right]$$ Work Exercises in order.
Use the method described here to find the inverse of $A=\left[\begin{array}{ll}4 & 2 \\ 7 & 3\end{array}\right].$

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Key Concepts

Matrix Inverse

The inverse of a matrix A, denoted as A?¹, is a matrix such that when it is multiplied by A, the result is the identity matrix. This concept is fundamental in linear algebra because it allows for the solution of systems of linear equations and the transformation of linear systems. An inverse exists only for square matrices that are non-singular, meaning they have a non-zero determinant.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. For a 2x2 matrix, the determinant is calculated as ad - bc. This value is crucial because it indicates whether the matrix has an inverse; a matrix is invertible if and only if its determinant is non-zero.

2x2 Matrix Inversion Formula

For a 2x2 matrix, the inverse is given by the formula where each element of the original matrix is rearranged and scaled by the reciprocal of the determinant. Specifically, for a matrix [a b; c d], the inverse is [d, -b; -c, a] divided by the determinant (ad - bc). This formula not only provides an efficient method for computing the inverse but also reinforces the role of the determinant in the existence of an inverse.

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