If I is a 2 × 2 identity matrix, find the determinant of the matrix.

Analysis

The question asks for the determinant of a $2 \times 2$ identity matrix, denoted as $I$.

Concept

An identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. For a $2 \times 2$ matrix, the identity matrix $I$ is defined as:

The determinant of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is calculated using the formula:

Solution

1. Identify the elements of the matrix: For the $2 \times 2$ identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$: • $a = 1$ • $b = 0$ • $c = 0$ • $d = 1$

2. Apply the determinant formula: $$\text{det}(I) = (1 \times 1) - (0 \times 0)$$ $$\text{det}(I) = 1 - 0$$ $$\text{det}(I) = 1$$

Conclusion

The determinant of the $2 \times 2$ identity matrix is 1.

Exam Tip

Always remember that the determinant of any identity matrix (whether $2 \times 2$, $3 \times 3$, or $n \times n$) is always 1. This is a useful shortcut that can save you time during your JAMB exam!