Matrix is a rectangular arrays of numbers used to represent data, transformations, and systems of equations. Three very important matrix concepts are the identity matrix, determinant, and inverse of a matrix. These ideas help us understand how matrices behave and how they are applied in problem-solving.

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1. Identity Matrix

What Is an Identity Matrix?

The identity matrix is a special square matrix that behaves like the number 1 in ordinary multiplication. When a matrix is multiplied by the identity matrix, the matrix remains unchanged.

Structure of an Identity Matrix

2 × 2 Identity Matrix

[ 1   0 ]
[ 0   1 ]

3 × 3 Identity Matrix

[ 1   0   0 ]
[ 0   1   0 ]
[ 0   0   1 ]

Key Property

If A is any square matrix of the same order:$$AI = IA = A$$

This makes the identity matrix important when defining the inverse of a matrix.

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2. Determinant of a Matrix

Meaning of the Determinant

The determinant is a single number calculated from a square matrix. It tells us:

• Whether a matrix has an inverse
• Whether a system of linear equations has a unique solution

Determinant of a 2 × 2 Matrix

For the matrix:$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

The determinant is:$$|A| = ad – bc$$

Visual Illustration

[ a   b ]   →   |A| = (a × d) − (b × c)
[ c   d ]

Example

$$A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$$$$|A| = (4 × 3) − (1 × 2) = 12 − 2 = 10$$

Interpretation

• If |A| ≠ 0, the matrix is non-singular (has an inverse)
• If |A| = 0, the matrix is singular (no inverse)

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3. Inverse of a Matrix

What Is an Inverse Matrix?

The inverse of a matrix reverses the effect of the original matrix.

If A⁻¹ is the inverse of A, then:$$AA^{-1} = A^{-1}A = I$$

Condition for an Inverse

A matrix has an inverse only if:$$|A| \neq 0$$

Inverse of a 2 × 2 Matrix

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

The inverse is:$$A^{-1} = \frac{1}{ad – bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

Step-by-Step Illustration

Original matrix:      Swap diagonals:      Change signs:
[ a   b ]             [ d   b ]            [ d  -b ]
[ c   d ]             [ c   a ]            [ -c  a ]

Multiply by 1 / determinant

Example

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

Determinant:$$|A| = (1 × 4) − (2 × 3) = 4 − 6 = -2$$

Inverse:$$A^{-1} = -\frac{1}{2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}$$summary of identity matrix, determinant and inverse

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Practice Questions

Section A: Identity Matrix

1. Write down the 2 × 2 identity matrix.
2. State the result of multiplying any matrix by the identity matrix.

Answers

[ 1   0 ]
[ 0   1 ]

2. The original matrix remains unchanged.

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Section B: Determinants

3. Find the determinant of:

$$\begin{pmatrix} 5 & 2 \\ 1 & 3 \end{pmatrix}$$

4. Determine whether the matrix is singular or non-singular:

$$\begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix}$$

Answers

$$|A| = (5 × 3) − (2 × 1) = 15 − 2 = 13$$

4. $$|A| = (2 × 2) − (4 × 1) = 4 − 4 = 0$$

The matrix is singular

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Section C: Inverse of a Matrix

5. Find the inverse of:

$$\begin{pmatrix} 3 & 1 \\ 2 & 1 \end{pmatrix}$$

Answer

Determinant:$$|A| = (3 × 1) − (1 × 2) = 1$$

Inverse:$$A^{-1} = \begin{pmatrix} 1 & -1 \\ -2 & 3 \end{pmatrix}$$

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Summary Diagram (Concept Link)

Determinant ≠ 0
       ↓
Matrix has an inverse
       ↓
Inverse × Original = Identity Matrix

Related topics

• Matrix-Math World