Introduction to Matrices and Matrix Operations

This lesson introduces matrices — rectangular arrays of numbers that are used to represent and solve systems of equations, describe geometric transformations, and model a wide range of mathematical and real-world problems. Matrices are a central topic in AQA A-Level Further Mathematics.

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What Is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. The order (or size) of a matrix with $m$m rows and $n$n columns is $m \times n$m×n.

$$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$$A=(14​25​36​)

This is a $2 \times 3$2×3 matrix (2 rows, 3 columns).

The entry in row $i$i, column $j$j is denoted $a_{ij}$aij​. For the matrix above, $a_{12} = 2$a12​=2 and $a_{23} = 6$a23​=6.

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Special Matrices

Matrix type	Description	Example
Square matrix	Same number of rows and columns ($n \times n$n×n)	$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$(13​24​)
Identity matrix $I$I	Square matrix with 1s on the diagonal and 0s elsewhere	$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$(10​01​)
Zero matrix $O$O	All entries are 0	$\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$(00​00​)
Diagonal matrix	Square; non-zero entries only on the main diagonal	$\begin{pmatrix} 3 & 0 \\ 0 & 7 \end{pmatrix}$(30​07​)
Symmetric matrix	$A = A^T$A=AT (entry $a_{ij} = a_{ji}$aij​=aji​)	$\begin{pmatrix} 1 & 5 \\ 5 & 2 \end{pmatrix}$(15​52​)
Column vector	$n \times 1$n×1 matrix	$\begin{pmatrix} 3 \\ -1 \end{pmatrix}$(3−1​)
Row vector	$1 \times n$1×n matrix	$\begin{pmatrix} 2 & 5 \end{pmatrix}$(2​5​)

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Matrix Addition and Subtraction

Two matrices can be added or subtracted only if they have the same order. The operation is performed element-by-element:

$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} + \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} a+e & b+f \\ c+g & d+h \end{pmatrix}$$(ac​bd​)+(eg​fh​)=(a+ec+g​b+fd+h​)

Worked Example: Compute $\begin{pmatrix} 3 & 1 \\ -2 & 4 \end{pmatrix} + \begin{pmatrix} 0 & -1 \\ 5 & 2 \end{pmatrix}$(3−2​14​)+(05​−12​).

$$= \begin{pmatrix} 3 & 0 \\ 3 & 6 \end{pmatrix}$$=(33​06​)

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Scalar Multiplication

Multiplying a matrix by a scalar $k$k multiplies every entry by $k$k:

$$k\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ka & kb \\ kc & kd \end{pmatrix}$$k(ac​bd​)=(kakc​kbkd​)

Example: $3\begin{pmatrix} 2 & -1 \\ 0 & 4 \end{pmatrix} = \begin{pmatrix} 6 & -3 \\ 0 & 12 \end{pmatrix}$3(20​−14​)=(60​−312​)

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The Transpose

The transpose of a matrix $A$A, written $A^T$AT, is obtained by swapping rows and columns:

If $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$A=(14​25​36​), then $A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$AT=​123​456​​

Properties:

• $(A^T)^T = A$(AT)T=A
• $(A + B)^T = A^T + B^T$(A+B)T=AT+BT
• $(kA)^T = kA^T$(kA)T=kAT
• $(AB)^T = B^T A^T$(AB)T=BTAT (note the reversal of order)

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Equality of Matrices

Two matrices $A$A and $B$B are equal if and only if:

1. They have the same order
2. Corresponding entries are equal: $a_{ij} = b_{ij}$aij​=bij​ for all $i, j$i,j

This is analogous to equating real and imaginary parts of complex numbers — a powerful technique for finding unknowns.

Worked Example: Find $x$x and $y$y if $\begin{pmatrix} 2x & 3 \\ y+1 & 5 \end{pmatrix} = \begin{pmatrix} 6 & 3 \\ 4 & 5 \end{pmatrix}$(2xy+1​35​)=(64​35​).

$2x = 6 \Rightarrow x = 3$2x=6⇒x=3; $y + 1 = 4 \Rightarrow y = 3$y+1=4⇒y=3.

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Properties of Matrix Addition

Property	Statement
Commutative	$A + B = B + A$A+B=B+A
Associative	$(A + B) + C = A + (B + C)$(A+B)+C=A+(B+C)
Additive identity	$A + O = A$A+O=A
Additive inverse	$A + (-A) = O$A+(−A)=O

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Practice Problems

1. Given $A = \begin{pmatrix} 1 & -2 \\ 3 & 0 \end{pmatrix}$A=(13​−20​) and $B = \begin{pmatrix} 4 & 1 \\ -1 & 5 \end{pmatrix}$B=(4−1​15​), compute $A + B$A+B, $A - B$A−B, and $3A$3A.
2. Find the transpose of $\begin{pmatrix} 2 & 5 \\ -1 & 3 \\ 0 & 7 \end{pmatrix}$​2−10​537​​.
3. Find $a$a and $b$b if $\begin{pmatrix} a + 1 & 2b \\ 3 & a - b \end{pmatrix} = \begin{pmatrix} 4 & 6 \\ 3 & 0 \end{pmatrix}$(a+13​2ba−b​)=(43​60​).

Solutions:

1. $A + B = \begin{pmatrix} 5 & -1 \\ 2 & 5 \end{pmatrix}$A+B=(52​−15​), $A - B = \begin{pmatrix} -3 & -3 \\ 4 & -5 \end{pmatrix}$A−B=(−34​−3−5​), $3A = \begin{pmatrix} 3 & -6 \\ 9 & 0 \end{pmatrix}$3A=(39​−60​)
2. $\begin{pmatrix} 2 & -1 & 0 \\ 5 & 3 & 7 \end{pmatrix}$(25​−13​07​)
3. $a + 1 = 4$a+1=4 gives $a = 3$a=3; $2b = 6$2b=6 gives $b = 3$b=3. Check: $a - b = 0$a−b=0 ✓

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Summary

• A matrix is a rectangular array; its order is rows × columns.
• Addition/subtraction: same-order matrices, element by element.
• Scalar multiplication: multiply every entry by the scalar.
• The transpose swaps rows and columns.
• The identity matrix $I$I and zero matrix $O$O play roles analogous to 1 and 0 in ordinary arithmetic.

Exam Tip: State the order of matrices clearly. Matrix operations require compatible dimensions — always check before proceeding.