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Definition

A matrix is a rectangular array of numbers arranged in rows and columns, used to represent and solve mathematical problems efficiently.

Before jumping into linear systems, like $A\vec{x} = \vec{b}$ , it's essential to understand how matrices behave and what operations we can perform on them.

Matrix Addition

You can add two matrices only if they have the same shape (same number of rows and columns).

Let:

A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}

Then:

A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}

Scalar Multiplication

You can also multiply a matrix by a scalar (single number):

k \cdot A = \begin{bmatrix} k a_{11} & k a_{12} \\ k a_{21} & k a_{22} \end{bmatrix}

Matrix Multiplication and Size Compatibility

Matrix multiplication is a row-by-column operation, not element-wise.

Rule: if matrix $A$ is of shape $(m \times n)$ and matrix $B$ is of shape $(n \times p)$ , then:

The multiplication $AB$ is valid;
The result will be a matrix of shape $(m \times p)$ .

Example:

Let:

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \ \ B = \begin{bmatrix} 5 \\ 6 \end{bmatrix}

$A$ is $(2 \times 2)$ and $B$ is $(2 \times 1)$ , then $AB$ is valid and results in a $(2 \times 1)$ matrix:

A \cdot B = \begin{bmatrix} 1 \cdot 5 + 2 \cdot 6 \\ 3 \cdot 5 + 4 \cdot 6 \end{bmatrix} = \begin{bmatrix} 17 \\ 39 \end{bmatrix}

Transpose of a Matrix

The transpose of a matrix flips rows and columns. It is denoted as $A^T$ .

Let:

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

Then:

A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}

Properties:

$(A^T)^T = A$ ;
$(A + B)^T = A^T + B^T$ ;
$(AB)^T = B^T A^T$ .

Determinant of a Matrix

2×2 Matrix

For:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

The determinant is:

\det(A) = ad - bc

3×3 Matrix

For:

A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}

The determinant is:

\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

This method is called cofactor expansion.

Larger matrices (4×4 and up) can be expanded recursively.
The determinant is useful because it indicates whether a matrix has an inverse (non-zero determinant).

Inverse of a Matrix

The inverse of a square matrix $A$ is denoted as $A^{-1}$ . It satisfies $A \cdot A^{-1} = I$ , where $I$ is the identity matrix.

Only square matrices with non-zero determinant have an inverse.

Example:

If matrix A is:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

Then its inverse matrix $A^{-1}$ is:

A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Where $\det(A) \neq 0$ .

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