Matrices & Determinants Reference Cheat Sheet

A printable reference covering matrix dimensions, operations, multiplication, identity matrices, determinants, inverses, and linear systems for grades 10-12.

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Matrices organize numbers into rows and columns so they can represent data, transformations, and systems of equations. This cheat sheet helps students remember the notation, rules, and formulas needed for common matrix and determinant problems. It is especially useful because matrix operations have strict size rules that are easy to mix up. A clear reference can prevent small notation mistakes from becoming wrong solutions. The main ideas include matrix order, addition, scalar multiplication, matrix multiplication, determinants, and inverses. For a $2 \times 2$ matrix, the determinant formula is $ad - bc$ , and it tells whether an inverse exists. Matrix multiplication uses row-by-column dot products, so in general $AB \neq BA$ . Inverses can be used to solve matrix equations and linear systems when the determinant is not zero.

Key Facts

A matrix with $m$ rows and $n$ columns has order $m \times n$ .
Matrices can be added or subtracted only when they have the same order, so $A + B$ is defined only if both matrices are $m \times n$ .
Scalar multiplication means multiplying every entry by the same number, so if $A = [a_{ij}]$ , then $kA = [ka_{ij}]$ .
Matrix multiplication $AB$ is defined only when the number of columns in $A$ equals the number of rows in $B$ .
If $A$ is $m \times n$ and $B$ is $n \times p$ , then $AB$ has order $m \times p$ .
For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , the determinant is $\det(A) = ad - bc$ .
For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ with $ad - bc \neq 0$ , the inverse is $A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ .
A square matrix $A$ has an inverse only if $\det(A) \neq 0$ .

Vocabulary

Matrix: A matrix is a rectangular array of numbers arranged in rows and columns.
Entry: An entry is one value inside a matrix, often written as $a_{ij}$ for the entry in row $i$ and column $j$ .
Order: The order of a matrix describes its size as rows by columns, written $m \times n$ .
Determinant: The determinant is a number found from a square matrix that helps decide whether the matrix has an inverse.
Identity Matrix: An identity matrix is a square matrix with $1$ s on the main diagonal and $0$ s elsewhere, and it acts like the number $1$ in multiplication.
Inverse Matrix: An inverse matrix $A^{-1}$ is a matrix that satisfies $AA^{-1} = A^{-1}A = I$ .

Common Mistakes to Avoid

Adding matrices with different orders is wrong because matrix addition requires matching positions in matrices of the same size.
Multiplying corresponding entries to find $AB$ is wrong because matrix multiplication uses row-by-column dot products, not entry-by-entry multiplication.
Assuming $AB = BA$ is wrong because matrix multiplication is usually not commutative and the two products may be different or not both defined.
Forgetting to check $\det(A) \neq 0$ before finding $A^{-1}$ is wrong because a matrix with determinant $0$ has no inverse.
Using the inverse formula without switching $a$ and $d$ or changing the signs of $b$ and $c$ is wrong because $A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ for a $2 \times 2$ matrix.

Practice Questions

1 Find $A + B$ for $A = \begin{bmatrix} 2 & -1 \\ 4 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 5 \\ -2 & 7 \end{bmatrix}$ .
2 Compute $AB$ if $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 5 \\ -1 & 2 \end{bmatrix}$ .
3 Find $\det(A)$ and decide whether $A^{-1}$ exists for $A = \begin{bmatrix} 6 & 2 \\ 9 & 3 \end{bmatrix}$ .
4 Explain why $AB$ may be undefined even when both $A$ and $B$ are valid matrices.

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