Welcome, dear student! In this lesson, we will learn about Matrices — one of the most important topics in Grade 11 Mathematics. Matrices are used everywhere: in computer graphics, solving equations, economics, and even in your phone’s camera filters! So let us understand them step by step. Ready? Let’s begin!

1. What is a Matrix?

A matrix (plural: matrices) is a rectangular arrangement of numbers, symbols, or expressions arranged in rows and columns. We always write matrices inside brackets — either square brackets \([ \ ]\) or parentheses \((\ )\).

Each number inside a matrix is called an element (or entry). We identify each element by its position — which row and which column it sits in.

Example: Consider the matrix:

This matrix has 2 rows and 3 columns. We say it is a “2 by 3” matrix, written as \(2 \times 3\).

• \(a_{11} = 3\) (row 1, column 1)
• \(a_{12} = 5\) (row 1, column 2)
• \(a_{23} = 4\) (row 2, column 3)

Think about it: if a matrix has \(m\) rows and \(n\) columns, its order (or size) is \(m \times n\). Can you tell the order of a matrix with 4 rows and 1 column? Yes — it is \(4 \times 1\). Good!

2. Types of Matrices

There are several special types of matrices you must know. Let us go through each one carefully with examples.

2.1 Row Matrix

A matrix that has exactly one row is called a row matrix. Its order is \(1 \times n\).

2.2 Column Matrix

A matrix that has exactly one column is called a column matrix. Its order is \(m \times 1\).

2.3 Square Matrix

A matrix in which the number of rows equals the number of columns is called a square matrix. Its order is \(n \times n\).

In a square matrix, the elements \(a_{11}, a_{22}, a_{33}, \ldots\) form the principal diagonal (or main diagonal).

2.4 Diagonal Matrix

A square matrix in which all elements outside the principal diagonal are zero is called a diagonal matrix.

2.5 Scalar Matrix

A diagonal matrix in which all the diagonal elements are equal is called a scalar matrix.

2.6 Identity Matrix (Unit Matrix)

A scalar matrix in which the diagonal elements are all 1 is called an identity matrix, denoted by \(I\).

The identity matrix acts like the number 1 in matrix multiplication — any matrix multiplied by \(I\) stays the same! We will see this later.

2.7 Zero Matrix (Null Matrix)

A matrix in which all elements are zero is called a zero matrix, denoted by \(O\).

2.8 Symmetric Matrix

A square matrix is symmetric if \(A = A^T\), meaning the element in row \(i\), column \(j\) equals the element in row \(j\), column \(i\). In other words, \(a_{ij} = a_{ji}\) for all \(i, j\).

Notice: \(a_{12} = 5 = a_{21}\), \(a_{13} = 2 = a_{31}\), \(a_{23} = 6 = a_{32}\). So this is symmetric!

2.9 Skew-Symmetric Matrix

A square matrix is skew-symmetric if \(A = -A^T\), meaning \(a_{ij} = -a_{ji}\) for all \(i, j\). This also means all diagonal elements must be zero (since \(a_{ii} = -a_{ii} \Rightarrow a_{ii} = 0\)).

3. Transpose of a Matrix

The transpose of a matrix \(A\), written as \(A^T\) (or \(A’\)), is obtained by interchanging rows and columns. That is, the first row becomes the first column, the second row becomes the second column, and so on.

If \(A\) has order \(m \times n\), then \(A^T\) has order \(n \times m\).

Worked Example:

Find the transpose of:

Solution: Interchange rows and columns:

See? The first row \((2, 5, 1)\) became the first column, and the second row \((3, 7, 4)\) became the second column.

4. Matrix Addition and Subtraction

Two matrices can be added or subtracted only if they have the same order (same number of rows and same number of columns). We add or subtract corresponding elements.

Worked Example 1:

Find \(A + B\) and \(A – B\).

Solution:

Worked Example 2:

Find \(A + B\).

Solution:

5. Scalar Multiplication of a Matrix

To multiply a matrix by a scalar (a regular number), we multiply every element of the matrix by that scalar.

Worked Example:

If \(A = \begin{pmatrix} 2 & -1 & 3 \\ 0 & 5 & 4 \end{pmatrix}\), find \(3A\) and \(-2A\).

Solution:

Have you noticed how scalar multiplication is like “distributing” the number to every element? It is straightforward, but be careful with negative signs — they affect every single element!

6. Matrix Multiplication

Matrix multiplication is one of the most important operations. It is NOT the same as multiplying element by element!

6.1 Condition for Multiplication

Two matrices \(A\) (of order \(m \times p\)) and \(B\) (of order \(p \times n\)) can be multiplied as \(AB\) only if the number of columns of \(A\) equals the number of rows of \(B\). The resulting matrix \(AB\) has order \(m \times n\).

6.2 How to Multiply

To find the element in row \(i\), column \(j\) of the product \(AB\): take the \(i\)-th row of \(A\) and the \(j\)-th column of \(B\), multiply corresponding elements, and add them up.

Worked Example 1:

Find \(AB\).

Solution:

\(AB\) has order \(2 \times 2\).

Worked Example 2: Now find \(BA\) and compare with \(AB\).

Worked Example 3: Multiplying a \(2 \times 3\) matrix by a \(3 \times 2\) matrix.

Solution: \(AB\) will be \(2 \times 2\).

6.3 Properties of Matrix Multiplication

• NOT commutative: \(AB \neq BA\) in general
• Associative: \((AB)C = A(BC)\)
• Distributive over addition: \(A(B + C) = AB + AC\)
• \(AI = IA = A\) (identity matrix acts like 1)
• \(AO = OA = O\) (zero matrix gives zero product)

7. Determinant of a Matrix

The determinant is a special number that we can calculate from a square matrix only. It tells us important things about the matrix — for example, whether the matrix has an inverse.

7.1 Determinant of a \(2 \times 2\) Matrix

For a \(2 \times 2\) matrix:

The determinant is:

Worked Example 1:

\(|A| = (3)(7) – (5)(2) = 21 – 10 = 11\)

Worked Example 2:

\(|B| = (4)(-3) – (-2)(6) = -12 + 12 = 0\)

When the determinant is zero, we say the matrix is singular (it has no inverse). When the determinant is non-zero, the matrix is non-singular (it has an inverse).

7.2 Determinant of a \(3 \times 3\) Matrix

For a \(3 \times 3\) matrix:

We expand along the first row:

Worked Example:

Solution:

7.3 Properties of Determinants

• \(|A^T| = |A|\)
• If two rows (or columns) are swapped, the determinant changes sign.
• If two rows (or columns) are identical, \(|A| = 0\).
• \(|kA| = k^n |A|\) where \(n\) is the order of the square matrix.
• \(|AB| = |A| \cdot |B|\)
• \(|A^{-1}| = \dfrac{1}{|A|}\) (when \(|A| \neq 0\))

8. Inverse of a Matrix

The inverse of a square matrix \(A\), denoted \(A^{-1}\), is a matrix such that:

An inverse exists only if \(|A| \neq 0\) (i.e., \(A\) is non-singular).

8.1 Inverse of a \(2 \times 2\) Matrix

For \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\):

Worked Example 1: Find the inverse of \(A = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}\).

Solution:

Step 1: \(|A| = (3)(4) – (1)(2) = 12 – 2 = 10 \neq 0\) ✓

Step 2: Swap diagonal: \(\begin{pmatrix} 4 & ? \\ ? & 3 \end{pmatrix}\)

Step 3: Negate off-diagonal: \(\begin{pmatrix} 4 & -1 \\ -2 & 3 \end{pmatrix}\)

Step 4: Divide by \(|A|\):

Verification:

8.2 Inverse of a \(3 \times 3\) Matrix

For a \(3 \times 3\) matrix, the process involves:

1. Find the determinant \(|A|\). If \(|A| = 0\), stop — no inverse exists.
2. Find the matrix of cofactors (each element replaced by its cofactor).
3. Take the transpose of the cofactor matrix to get the adjoint (also called adjugate), written as \(\text{adj}(A)\).
4. Apply the formula: \(A^{-1} = \dfrac{1}{|A|} \cdot \text{adj}(A)\)

Worked Example: Find the inverse of \(A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 2 & 3 & 4 \end{pmatrix}\).

Solution:

Step 1: Find \(|A|\):

Step 2: Find cofactors for each element:

Cofactor matrix: \(\begin{pmatrix} 1 & 2 & -2 \\ 1 & -2 & 1 \\ -1 & -1 & 1 \end{pmatrix}\)

Step 3: Transpose to get adjoint:

Step 4: Apply formula:

9. Solving Systems of Linear Equations Using Matrices

One of the most important applications of matrices is solving systems of linear equations. Let us see how!

9.1 Matrix Form of a System

A system of linear equations can be written as \(AX = B\), where:

• \(A\) = coefficient matrix
• \(X\) = column matrix of variables
• \(B\) = column matrix of constants

Example: Consider the system:

In matrix form:

So \(AX = B\), where \(A = \begin{pmatrix} 2 & 3 \\ 1 & -1 \end{pmatrix}\), \(X = \begin{pmatrix} x \\ y \end{pmatrix}\), \(B = \begin{pmatrix} 8 \\ 1 \end{pmatrix}\).

9.2 Solving Using the Inverse Method

If \(A\) is non-singular (\(|A| \neq 0\)), then:

Worked Example: Solve the system above using the inverse method.

Solution:

Step 1: \(|A| = (2)(-1) – (3)(1) = -2 – 3 = -5 \neq 0\) ✓

Step 2: Find \(A^{-1}\):

Step 3: \(X = A^{-1}B\):

So \(x = \dfrac{11}{5}\) and \(y = \dfrac{6}{5}\).

9.3 What if \(|A| = 0\)?

If \(|A| = 0\), the system either has:

• No solution (inconsistent system), or
• Infinitely many solutions (dependent system)

You cannot use the inverse method in this case. You would need other methods (like row reduction) to determine which case applies.

9.4 Cramer’s Rule (Alternative Method)

For a \(2 \times 2\) system \(AX = B\):

where \(A_x\) is matrix \(A\) with the first column replaced by \(B\), and \(A_y\) is \(A\) with the second column replaced by \(B\).

Example: Solve \(2x + 3y = 8\), \(x – y = 1\) using Cramer’s Rule.

Same answer! Both methods give \(x = \frac{11}{5}\), \(y = \frac{6}{5}\). ✓