Key matrices concepts include identity, determinant, inverse, and singular matrices, which help describe matrix properties and transformations.

Matrices are important tools in mathematics and science. They are used to solve systems of equations, describe transformations, and model real-life problems such as economics, physics, and computer graphics.
In this lesson, we will learn four important concepts related to square matrices:

• Identity Matrix
• Determinant of a Matrix
• Inverse of a Matrix
• Singular Matrix

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2. Identity Matrices

Definition

An identity matrix is a square matrix that has:

• 1’s on the main diagonal
• 0’s everywhere else

It is similar to the number 1 in multiplication, because multiplying any matrix by the identity matrix gives the same matrix.

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Notation of matrices

The identity matrix is denoted by I.

2 × 2 Identity Matrix

$$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

3 × 3 Identity Matrix

$$I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$I3​=​100​010​001​​

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Illustration

Main diagonal →   1   1   1
Other entries →   0   0   0

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Property

$$A \times I = I \times A = A$$

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3. Determinant of a Matrix

Definition

The determinant of a square matrix is a single number that tells us important information about the matrix, such as:

• Whether the matrix has an inverse
• Whether the system of equations has a unique solution
• The scaling factor of a geometric transformation

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Determinant of a 2 × 2 Matrix

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

The determinant is:$$|A| = ad – bc$$

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Example

$$A = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$$A=[31​24​] $$|A| = (3 \times 4) – (2 \times 1) = 12 – 2 = 10$$∣A∣=(3×4)−(2×1)=12−2=10

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The determinant represents area scaling:

• If |A| = 2 → area doubles
• If |A| = 0 → area collapses to a line

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4. Inverse of a Matrices

Definition

The inverse of a matrix A is another matrix $A^{-1}$A−1 such that:$$A \times A^{-1} = I$$

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Inverse of a 2 × 2 Matrix

For:$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$$$A^{-1} = \frac{1}{ad – bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

Note: The inverse exists only if the determinant is not zero.

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Example

$$A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$$

Step 1: Find determinant:$$|A| = (2 \times 3) – (1 \times 5) = 6 – 5 = 1$$

Step 2: Find inverse:$$A^{-1} = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$$

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Illustration

Matrix A → changes a vector  
Inverse A⁻¹ → returns it back to original

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5. Singular Matrix

Definition

A singular matrix is a matrix that has no inverse.

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Condition

A matrix is singular if: ∣A∣=0

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Example

$$A = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix}$$A=[21​42​] $$|A| = (2 \times 2) – (4 \times 1) = 4 – 4 = 0$$∣A∣=(2×2)−(4×1)=4−4=0

Therefore, A is singular and cannot be inverted.

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Illustration

Transformation squashes shape into a line → no way to reverse it

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6. Summary Table

Concept	Meaning	Key Condition
Identity Matrix	Matrix with 1’s on diagonal	Acts like number 1
Determinant	Single number describing matrix properties	The matrix is not a zero matrix
Inverse Matrix	Matrix that reverses A	Exists if A exists
Singular Matrix	Matrix without inverse

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7. Practice Questions

Question 1

Find the determinant of:$$A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$$

Answer:
$|A| = (4×6) − (7×2) = 24 − 14 = 10$

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Question 2

Write the 3 × 3 identity matrix.

Answer:$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

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Question 3

Determine whether the matrix is singular:$$A = \begin{bmatrix} 1 & 3 \\ 2 & 6 \end{bmatrix}$$

Answer:
$|A| = (1×6) − (3×2) = 6 − 6 = 0$

Matrix is singular.

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Question 4

Find the inverse of:$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$

Answer:$$|A| = (1×4) − (2×3) = 4 − 6 = -2$$$$A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}$$

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8. Conclusion

Understanding identity, determinant, inverse, and singular matrices is important in solving systems of equations, physics, engineering, and computer science. These concepts help us understand when a matrix can be reversed and how it transforms.

Related Topics

• Operation on matrices
• Introduction to matrices
•  MATRICES AND TRANSFORMATIONS

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