Determinant

Study Material (PDF)
Self Assesment
Study Material (PDF)
Self Assesment

Click on each question to see answer & solution.

Question No: #11

If `A = [a_{ij}]_{n\timesn}` and `|A| = d \ne0`, then `|\adj\ A|` is -- (a) `d` (b) `d^n` (c) `d^{n-1}` (d) `d^{n+1}`

Question No: #12

If `A` is an orthogonal matrix of order 3, then value of `|A|` is -- (a) `0` (b) `1` (c) `-1` (d) `\pm1`

Question No: #13

The determinant $ \begin{vmatrix} a & b & a\alpha+b \\ b & c & b\alpha+c \\ a\alpha+b & b\alpha+c & 0 \end{vmatrix} = 0 $ if `a, b, c` are in -- (a) A.P. (b) G.P. (c) H.P. (d) None of these.

Question No: #14

Prove that $ \begin{vmatrix} a & b & a\alpha+b\beta \\ b & c & b\alpha+c\beta \\ a\alpha+b\beta & b\alpha+c\beta & 0 \end{vmatrix} = (b^2-ac) (a\alpha^2+2b\alpha\beta+c\beta^2) $

Question No: #15

Without expanding, prove that: $ \begin{vmatrix} a+bx & c+dx & p+qx \\ ax+b & cx+d & px+q \\ u & v & w \end{vmatrix} = (1-x^2) \begin{vmatrix} a & c & p \\ b & d & q \\ u & v & w \end{vmatrix} $

Question No: #16

If $ \begin{vmatrix} -5 & 5 & 10 \\ 5 & -5 & x \\ 0 & 10 & 5 \end{vmatrix} =0 $ then find `x`
Ans: `-10`
SEE SOLUTION

Question No: #17

Solve: $ \begin{vmatrix} x+a & b & c \\ b & x+c & a \\ c & a & x+b \end{vmatrix} = 0 $
Ans: `x=-(a+b+c), \pm\frac{1}{\sqrt{2}}\sqrt{(a-b)^2+(b-c)^2+(c-a)^2}`
SEE SOLUTION

Question No: #18

Solve : $ \begin{vmatrix} x+4 & 3 & 5 \\ 3 & x+4 & 5 \\ 3 & 5 & x+4 \end{vmatrix} = 0 $
Ans: `x = \pm1, -12`
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Question No: #19

Solve for `x`. $ \begin{vmatrix} x+2 & 3 & 3 \\ 3 & x+4 & 5 \\ 3 & 5 & x+4 \end{vmatrix} = 0 $
Ans: 0, 1, -11
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Question No: #20

Solve: $ \begin{vmatrix} 2-x & 2 & 3 \\ 2 & 5-x & 6 \\ 3 & 4 & 10-x \end{vmatrix} = 0 $
Ans: `1, 8\pm \sqrt{37}`
SEE SOLUTION