6.08 Inverse matrices

In mathematics, when you multiply a number by its inverse the result is $1$1. In other words, a number's inverse is the unique number which results in $1$1 when the two are multiplied together. In matrices, the product of a matrix and its inverse (if it exists) is the identity matrix, $I$I.

With real numbers, a number's inverse is its reciprocal. For instance, $6\times\frac{1}{6}=1$6×16​=1, where $\frac{1}{6}$16​ and $6$6 are inverses of one another. The concept of an inverse is often used when solving equations, like solving $5x=20$5x=20.

Notation

The inverse of $6$6 can also be written as $6^{-1}$6−1. As with this example, we have that:

$A^{-1}$A−1 represents the inverse of matrix $A$A.

The determinant of a $2\times2$2×2 matrix

The determinant is a number that is used to find the inverse of a matrix (if it exists). If the determinant is zero then the inverse is said to be undefined, this can be useful when solving systems of linear equations later.

Generally, the determinant of a matrix $A$A is written as $det(A)$det(A) and is calculated as follows:

Practice question

Question 1

The inverse of a 2 x 2 matrix

To find the inverse of a $2\times2$2×2 matrix, say $A$A, we swap the entries along the main-diagonal, and multiply the entries in the off-diagonal by $-1$−1. Then we multiply the result by $\frac{1}{det(A)}$1det(A)​.

The inverse matrix $A^{-1}$A−1 has the property that when we multiply it by $A$A, we get the identity matrix.

$AA^{-1}=A^{-1}A=I$AA−1=A−1A=I

Practice questions

Question 2

Question 3

Question 4