Question

(a) Find the value of k if the matrix [ 2 -1 4 5 7 k 2 8 9]
is singular.

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:A = \begin{bmatrix}</p><p>2 & - 1 & 4 \\</p><p>5 & 7 & k \\</p><p>2 &8 & 9 \\</p><p>\end{bmatrix}$

It is given that

$\rm :\longmapsto\:A \: is \: singular \: matrix.$

$\rm :\implies\: |A| = 0$

$\rm :\longmapsto\:\begin{array}{|ccc|}</p><p>2 & - 1 & 4 \\</p><p>5 & 7 & k \\</p><p>2 &8 & 9 \\</p><p>\end{array} = 0$

$\rm :\longmapsto\:2\begin{array}{|cc|}\sf 7 &\sf k \\ \sf 8 &\sf 9 \\\end{array} + 1\begin{array}{|cc|}\sf 5 &\sf k \\ \sf 2 &\sf 9 \\\end{array} + 4\begin{array}{|cc|}\sf 5 &\sf 7 \\ \sf 2 &\sf 8 \\\end{array} = 0$

$\rm :\longmapsto\:2(63 - 8k) + (45 - 2k) + 4(40 - 14) = 0$

$\rm :\longmapsto\:126 - 16k + 45 - 2k + 104 = 0$

$\rm :\longmapsto\:275 - 18k = 0$

$\rm :\implies\:k = \dfrac{275}{18}$

Additional Information :-

• 1. The determinant remains unaltered if its rows are changed into columns and the columns into rows. This is known as the property of reflection.

• 2. If all the elements of a row (or column) are zero, then the determinant is zero.

• 3. If the all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant is zero.

• 4. The interchange of any two rows (or columns) of the determinant changes its sign.

• 5. If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements