2-2 Matrix Inverse

2.2 The Inverse of a Matrix Matrix inverses are essential in Linear Algebra. You will need to know the inverse formula for a 2 x 2 matrix for later exercises. This only works if the determinant is not zero. To find the inverse of a 2 x 2 matrix, interchange the diagonal entries, change the signs of the off-diagonal entries. Then multiply by the reciprocal of the determinant.  a b  c d  

1



1  d b  ad  bc  c a 

The proof of Theorem 5 is important. Recall that the phrase “has a unique solution” includes both existence and uniqueness. Unless A is a 2 x 2, Theorem 5 is not usually used. It is faster to row reduce [A b]. Since matrix multiplication is not commutative, DO NOT write A/B. This could mean either A-1B or AB-1. Each elementary row operation amounts to left multiplication by an elementary matrix. So, for A  U, then F, a product of elementary matrices, exists such that FA = U.