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Matrices

Matrix operations, determinants and inverses.

A matrix is a rectangular array of numbers that represents a linear transformation or a system of equations. Matrices of the same size add entry by entry. Multiplying $AB$ takes dot products of rows of $A$ with columns of $B$ , so the number of columns of $A$ must equal the number of rows of $B$ , and in general $AB \ne BA$ .

The determinant of a square matrix measures how the transformation scales area or volume; a determinant of zero means the matrix squashes space flat and has no inverse. When the inverse exists, the system $A\mathbf{x} = \mathbf{b}$ has the unique solution $\mathbf{x} = A^{-1}\mathbf{b}$ .

Key ideas

For $2 \times 2$ : $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$ .
$2 \times 2$ inverse: $\frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ .
Row reduction solves systems of any size and finds rank and inverses.
The identity matrix $I$ satisfies $AI = IA = A$ .

Worked example

Solve $\begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}\mathbf{x} = \begin{pmatrix} 4 \\ 11 \end{pmatrix}$ .

$\det = 2 \cdot 3 - 1 \cdot 5 = 1$ .
$A^{-1} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$ .
$\mathbf{x} = A^{-1}\mathbf{b} = \begin{pmatrix} 12 - 11 \\ -20 + 22 \end{pmatrix}$ .
$\mathbf{x} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ .

Answer: $x = 1$ , $y = 2$

Common mistakes

Multiplying matrices entry by entry.
Assuming $AB = BA$ .
Swapping the wrong entries in the $2 \times 2$ inverse formula.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Find $\det\begin{pmatrix} 4 & 3 \\ 2 & 5 \end{pmatrix}$ .
Show answer
$14$
2.Find the top-left entry of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}^2$ .
Show answer
$7$
3.For which $k$ is $\begin{pmatrix} 2 & k \\ 3 & 6 \end{pmatrix}$ not invertible?
Show answer
$k = 4$

Matrices

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