Determine whether an inverse exists and calculate it

A determinant is a scalar, not a matrix

For the matrix shown, subtract the other-diagonal product from the main-diagonal product. The main-diagonal product is reduced by the other-diagonal product

Given matrix

\(\displaystyle A=\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}\)

Find det(A) for the matrix A

Given matrix

\(\displaystyle S=\begin{bmatrix} 2 & 6 \\ 1 & 3 \end{bmatrix}\)

Find det(S) for the matrix S

An inverse reverses multiplication on both sides

For a square A, its inverse is a matrix B with AB=BA=I. A 2×2 matrix has an inverse exactly when ad−bc≠0. Swap the diagonal entries, negate the off-diagonal entries, then divide the entire result by the determinant

Given matrix

\(\displaystyle A=\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}\)

For the matrix A, find the entry in row 1, column 2 of A⁻¹

What follows when a 2×2 matrix has determinant zero?

Derive the inverse formula

Multiply A by D in both orders; explain why division by ad−bc is permitted only when it is nonzero

Model: an adjugate product

Direct multiplication gives AD=DA=(ad−bc)I. When ad−bc≠0, B=D/(ad−bc) therefore satisfies AB=BA=I. When the determinant is zero the division is undefined; if the first row (a,b) is nonzero, v=(b,−a)ᵀ is nonzero and Av=(0,bc−ad)ᵀ=0. If that row is zero but (c,d) is nonzero, use v=(d,−c)ᵀ; for the zero matrix any nonzero v works. An inverse would imply v=A⁻¹Av=0, a contradiction