4.11 The Inverse and Determinant of a Matrix

Matrix Inverses and Determinants

For AP Precalculus, the inverse of a 2 x 2 matrix exists exactly when its determinant is not zero. The determinant of $A = [a \ b; c \ d]$ is $ad - bc$, and that value tells you whether the matrix is invertible, whether two row or column vectors are parallel, and the area of the parallelogram those vectors span.

The identity matrix $I_2$ is the matrix version of multiplying by 1. If $A^{-1}$ exists, then $AA^{-1} = A^{-1}A = I$.

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Identity Matrix and Inverses

The identity matrix, denoted as I, is a special type of square matrix that has the property that when it is multiplied by any other matrix of the same size, the resulting matrix is equal to the original matrix.

The identity matrix has the size of n x n, where n is the number of rows and columns. The identity matrix has ones on the diagonal from the top left to bottom right, and zeros everywhere else. This diagonal of ones is often referred to as the main diagonal. The identity matrix is also known as the unit matrix.

The product of a square matrix and its inverse, when it exists, is the identity matrix of the same size. This means that if $A$ is a square matrix and $A^-1$ is the inverse of $A$, then $A * A^-1 = A^-1 * A = I$, where $I$ is the identity matrix of the same size as $A$.

The inverse of a 2 x 2 matrix, when it exists, can be calculated both with and without technology.

One way to calculate the inverse of a 2 x 2 matrix is to use the formula:

If $A = [a11, a12; a21, a22]$, then $A^-1 = \frac{1}{a11*a22 - a12*a21} * [a22, -a12; -a21, a11]$, where $det(A) = a11*a22 - a12*a21$.

Determinants

The determinant of a matrix is a scalar value that can be calculated for square matrices, and it is denoted as $det(A)$. The determinant of a matrix A is a measure of its invertibility and it can be used to find the inverse of a matrix, when it exists.

For a 2 x 2 matrix A (see image above), the determinant is calculated as the scalar value $ad - bc$. This is a simple calculation that can be done by hand or with the help of a calculator.

The determinant of a 2×2 matrix can be calculated with or without technology (e.g., on a calculator).

Optional/Not required for AP Precalculus: Determinants of larger matrices (e.g., 3×3) exist but are not assessed in this unit.

Relating Determinants and Parallel Vectors

If a 2 x 2 matrix $A = [v1 \ v2]$ consists of two column vectors $v1$ and $v2$ from R2, then $|det(A)| = |v1|*|v2|*|sin(θ)|$, and this absolute value equals the area of the parallelogram spanned by $v1$ and $v2$. If $det(A) = 0$, the vectors are parallel. (The sign of $det(A)$ indicates orientation, depending on the order of $v1, v2$.)

It's worth noting that the same holds true for a 2 x 2 matrix $A = [v1; v2]$ if the matrix consists of two row vectors $v1$ and $v2$ from R2: the area is $|det(A)|$ and if the determinant equals 0, then the vectors are parallel.

Invertibility Condition

The square matrix A has an inverse if and only if its determinant, denoted as $det(A)$, is not equal to 0 [$det(A) =/= 0$]. This means that a square matrix A is invertible if and only if its determinant is non-zero. This is often referred to as the invertibility condition of a matrix.