# Find the determinant of the matrix $\begin{vmatrix} 2 & 7 & 37\\ 3& 6 & 33\\ 4 & 5 & 29 \end{vmatrix}$

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Find the determinant of the matrix $\begin{vmatrix} 2 & 7 & 37\\ 3& 6 & 33\\ 4 & 5 & 29 \end{vmatrix}$
1. 234
2. 132
3. 83
4. 0
5. None of these

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Correct Answer - Option 4 : 0

Concept:

Properties of Determinant of a Matrix:

• If each entry in any row or column of a determinant is 0, then the value of the determinant is zero.
• For any square matrix say A, |A| = |AT|.
• If we interchange any two rows (columns) of a matrix, the determinant is multiplied by -1.
• If any two rows (columns) of a matrix are same then the value of the determinant is zero.

Calculation:

$\begin{vmatrix} 2 & 7 & 37\\ 3& 6 & 33\\ 4 & 5 & 29 \end{vmatrix}$

Apply C2 → 5C2 + C1, we get

$\begin{vmatrix} 2 & 37 & 37\\ 3& 33 & 33\\ 4 & 29 & 29 \end{vmatrix}$

As we can see that the second and the third column of the given matrix are equal.

We know that, if any two rows (columns) of a matrix are same then the value of the determinant is zero.

∴ $\begin{vmatrix} 2 & 7 & 37\\ 3& 6 & 33\\ 4 & 5 & 29 \end{vmatrix}$ = 0