Expansion of Determinants

1 Mark Questions

1. If <math>\begin{vmatrix} 2x & 5 \\ 8 & x \end{vmatrix} = \begin{vmatrix} 6 & -2 \\ 7 & 3 \end{vmatrix}</math>, then write the value of

Χ. Delhi 2014

Firstly, expand both determinants, which gives equation in x and then solve that equation for finding the value of x.

Given, <math>\begin{vmatrix} 2x & 5 \\ 8 & x \end{vmatrix} = \begin{vmatrix} 6 & -2 \\ 7 & 3 \end{vmatrix}</math> <math>2x^2 - 40 = 18 - (-14)</math> <math>2x^2 - 40 = 18 + 14</math> <math>\Rightarrow</math> <math>2x^2 - 40 = 32</math> <math>\Rightarrow</math> <math>2x^2 = 32 + 40</math> <math>\Rightarrow</math> <math display="block">2x^2 = 72 \implies x^2 = 36</math>

• ٠. <math>x = \pm 6</math>

$$(1)

1. If <math>\begin{vmatrix} 3x & 7 \\ -2 & 4 \end{vmatrix} = \begin{vmatrix} 8 & 7 \\ 6 & 4 \end{vmatrix}</math>, then find the value of x. All India 2014

Given, <math>\begin{vmatrix} 3x & 7 \\ -2 & 4 \end{vmatrix} = \begin{vmatrix} 8 & 7 \\ 6 & 4 \end{vmatrix}</math>

On expanding the determinant of both sides, we get

<math>3x \times 4 - (-2) \times 7 = 8 \times 4 - 6 \times 7</math>

<math>12x - (-14) = 32 - 42</math>

<math>\Rightarrow</math>

<math>12x + 14 = -10</math>

<math>\Rightarrow</math> <math>\Rightarrow</math>

<math>12x = -10 - 14</math>

<math display="block">12x = -24 \implies x = -\frac{24}{12}</math>

<math>x = -2</math> (1)

1. If A is a <math>3 \times 3</math> matrix, <math>|A| \neq 0</math> and <math>|3A| = k|A|</math>, then

write the value of k. Foreign 2014

We know that, if A is a square matrix of order n. Then, <math>|kA| = k^n |A|</math>.

Here, the matrix A is of order 3.

<math>|3A| = (3)^3 |A| = 27 |A|</math> ٠. On comparing with <math>k|A|</math>, we get

<math>k = 27</math> (1)