Let A be the 2 ×2 matrix A=[ a b c d ] Show that if a d-b c ≠0 , then 𝐀^-1=[ (d)

step 1 So we start with the matrix A given by A, B, C, D. Now we recall that A inverse is by definition

Let A be the 2 ×2 matrix A=[ a b c d ] Show that...

Let $A$ be the $2 \times 2$ matrix $$ A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right] $$ Show that if $a d-b c \neq 0$ , then $$ \mathbf{A}^{-1}=\left[\begin{array}{cc}{\frac{d}{a d-b c}} & {\frac{-b}{a d-b c}} \\ {\frac{-c}{a d-b c}} & {\frac{a}{a d-b c}}\end{array}\right] $$

Let $A$ be the $2 \times 2$ matrix

$$
A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]
$$
Show that if $a d-b c \neq 0$ , then

$$
\mathbf{A}^{-1}=\left[\begin{array}{cc}{\frac{d}{a d-b c}} & {\frac{-b}{a d-b c}} \\ {\frac{-c}{a d-b c}} & {\frac{a}{a d-b c}}\end{array}\right]
$$

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Key Concepts

Matrix Inversion

Matrix inversion refers to the process of finding a matrix, called the inverse, that when multiplied by the original matrix results in the identity matrix. This concept is fundamental in linear algebra since the inverse of a matrix, if it exists, provides a means to solve systems of linear equations, among other applications. It is essential that the original matrix has an inverse, which is true if and only if the matrix is non-singular (i.e., its determinant is not zero).

Determinant

The determinant is a scalar value computed from the elements of a square matrix that provides important information about the matrix, including whether it is invertible. For a 2x2 matrix, the determinant is calculated as ad - bc. A non-zero determinant confirms that the matrix is invertible, whereas a determinant equal to zero indicates that the matrix is singular and does not have an inverse.

Adjugate Matrix

The adjugate matrix (or adjoint) is formed by taking the matrix of cofactors, transposing it, and is used in the formula to compute the inverse of a matrix. For a 2x2 matrix, this involves swapping the elements along the main diagonal and changing the signs of the off-diagonal elements. Dividing the adjugate matrix by the determinant gives the inverse, providing an explicit formula for matrix inversion when the determinant is non-zero.

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