22. The determinant of the matrix is __________.

Introduction

Finding the determinant of the given matrix is an important topic in matrices for board exams
and competitive examinations. In this article, we will solve the determinant step by step, explain each
option clearly, and use shortcut methods wherever possible.

Question

Find the determinant of the matrix:

| 3   0   0 |
| 2   5   0 |
| 6   −8   −4 |

Step-by-Step Solution

Step 1: Observe the Matrix

The given matrix contains zeros in the third column. This indicates that the matrix is almost
upper triangular.

Step 2: Property of Determinants

For an upper triangular or lower triangular matrix, the determinant is equal to the
product of the diagonal elements.

Step 3: Identify Diagonal Elements

• a₁₁ = 3
• a₂₂ = 5
• a₃₃ = −4

Step 4: Calculate the Determinant

Determinant = 3 × 5 × (−4)

Determinant = −60

Final Answer

−60

Explanation of All Options

Option A: 0 ❌

The determinant becomes zero only if an entire row or column is zero or if two rows are identical.
This is not the case here.

Option B: 60 ❌

This ignores the negative diagonal element. Since one diagonal value is −4, the determinant must be negative.

Option C: −60 ✅

This option correctly multiplies all diagonal elements and accounts for the negative sign.

Option D: −20 ❌

This is a result of incorrect or incomplete multiplication and does not follow determinant rules.

Important Points to Remember

• If a matrix is triangular, determinant = product of diagonal elements.
• Zeros outside the diagonal do not affect the determinant value.
• Always check for shortcut properties before expanding.

Conclusion

The determinant of the given matrix is −60. Recognizing the triangular
structure helps solve the problem quickly and accurately, especially in competitive exams.