Evaluate \(\begin{vmatrix} a+ib & c+id \\ c-id & a-ib \end{vmatrix}\)

1

(a)

(a + ib) (a - ib) + (c + id) (c - id) = a²+b²+c²+d²

Write the value of determinant \(\begin{vmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{vmatrix}\).

1

(a)

0

What positive value of x makes the following pair of determinants equal?
\(\begin{vmatrix} 2x & 3 \\ 5 & x \end{vmatrix},\begin{vmatrix} 16 & 3 \\ 5 & 2 \end{vmatrix}\)

1

(a)

\( 2 x^{2}-15 =32-15 \Rightarrow x^{2}=16 \)
\(\Rightarrow x =\pm 4 \Rightarrow x=4(>0)\)

If A=\(\begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}\) write A^-1 in terms of A .

1

(a)

\(A^{-1}=\frac{1}{|A|} \text { adj } A \)
\(|A|=\left[\begin{array}{lr} 2 & 3 \\ 5 & -2 \end{array}\right]=-4-15=-19 \)
\(\Rightarrow A^{-1}=-\frac{1}{19}\left[\begin{array}{rr} -2 & -3 \\ -5 & 2 \end{array}\right]=\frac{1}{19}\left[\begin{array}{cc} 2 & 3 \\ 5 & -2 \end{array}\right]=\frac{1}{19} A\)

Find the value of p, such that the matrix \(\begin{bmatrix} -1 & 2 \\ 4 & p \end{bmatrix}\) is singular.

1

(a)

p=-8

Given I₂. Find |I₂| . Also find |3 I₂|.

1

(a)

9

Find the value of X, such that the points (0, 2), (1, x) and (3, 1) are collinear.

1

(a)

If points are collinear then area of triangle = 0.
\(\Rightarrow \frac{1}{2}\left|\begin{array}{ccc} 0 & 2 & 1 \\ 1 & x & 1 \\ 3 & 1 & 1 \end{array}\right|=0 \)
\(\Rightarrow \frac{1}{2}[0-2(1-3)+1(1-3 x)]=0 \)
\(\Rightarrow 4+1-3 x=0 \Rightarrow x=\frac{5}{3} \)

If for matrix A,|A|=3 find |5A|, where matrix A is order 2x2.

1

(a)

75

Find value of x, if \(\begin{vmatrix} 2 & 3 \\ x & 1 \end{vmatrix}=\begin{vmatrix} x & 3 \\ 2x & 5 \end{vmatrix}\)

1

(a)

10 - 12 = 5x - 6x \(\Rightarrow\) x = 2