Q1: Evaluate the determinants.
\(\begin{vmatrix} 2 & 4\\ -5 & -1 \end{vmatrix}\)
Ans:
\(\begin{vmatrix} 2 & 4\\ -5 & -1 \end{vmatrix}\) = 2 ( -1 ) – 4 ( -5 ) = – 2 + 20 = 18
Q2: Evaluate the determinant.
(i) \(\begin{vmatrix} cos \Theta & -sin \Theta \\ sin \Theta & cos \Theta \end{vmatrix}\\\)
(ii) \(\\\begin{vmatrix} x^{ 2 } – x + 1 & x – 1 \\ x + 1 & x + 1 \end{vmatrix}\)
Ans:
(i) (cos Ɵ )(cos Ɵ ) – ( -sin Ɵ ) (sin Ɵ ) = cos2 Ɵ + sin2 Ɵ = 1
(ii) = ( x2 – x + 1 ) ( x + 1 ) – ( x – 1 ) ( x + 1 )
= x3 – x2 + x + x2 – x + 1 – ( x2 – 1 )
= x3 + 1 – x2 + 1
= x3 – x2 + 2
Q3: If A = \(\begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}\) , then show that |2A| = 4 |A|
Ans: The given matrix is A = \(\begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}\)
Therefore, 2A = \(2 \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 8 & 4 \end{bmatrix}\)
Therefore, L.H.S. = |2A| = \(\begin{vmatrix} 2 & 4 \\ 8 & 4 \end{vmatrix}\) = 2 × 4 – 4 × 8 = 8 – 32 = -24
Now, |A| = \(\begin{vmatrix} 1 & 2 \\ 4 & 2 \end{vmatrix}\) =
1 × 2 – 2 × 4 = 2 – 8 = -6
Therefore, R.H.S. = 4|A| = 4 × ( -6) = -24
Therefore, L.H.S. = R.H.S.
Q.4: If A = \(\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}\) , then show that |3A| = 27|A|.
Ans: The given matrix is A = \(\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}\)
It can be observed that in the first column, two entries are zero. Thus, we expand along the first column (C1) for easier calculation.
|A| = \(1 \begin{vmatrix} 1 & 2 \\ 0 & 4 \end{vmatrix} – 0 \begin{vmatrix} 0 & 1 \\ 0 & 4 \end{vmatrix} + 0 \begin{vmatrix} 0 & 1 \\ 1 & 2 \end{vmatrix}\) = 1 ( 4 – 0 ) – 0 + 0 = 4
Therefore, 27|A| = 27 (4) = 108 …(i)
Now, 3A = \(3 \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 0 & 3 \\ 0 & 3 & 6 \\ 0 & 0 & 12 \end{bmatrix}\)
Therefore, |3A| = \(3 \begin{vmatrix} 3 & 6 \\ 0 & 12 \end{vmatrix} – 0 \begin{vmatrix} 0 & 3 \\ 0 & 12 \end{vmatrix} + 0 \begin{vmatrix} 0 & 3 \\ 3 & 6 \end{vmatrix}\)
= 3 ( 36 – 0 ) = 3 ( 36 ) = 108 …(ii)
From equations (i) and (ii), we have:
|3A| = 27|A|
Hence, the given result is proved.
Q5: Evaluate the determinants
(i) \(\begin{vmatrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}\\\)
(ii) \(\begin{vmatrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{vmatrix}\\\)
(iii) \(\begin{vmatrix} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{vmatrix}\\\)
(iv) \(\begin{bmatrix} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{bmatrix}\)
Ans:
(i) Let A = \(\begin{vmatrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}\)
It can be observed that in the second row, two entries are zero. Thus, we expand along the second row for easier calculations.
|A| = \(0 \begin{vmatrix} -1 & -2 \\ -5 & 0 \end{vmatrix} + 0 \begin{vmatrix} 3 & -2 \\ 3 & 0 \end{vmatrix} – (-1) \begin{vmatrix} 3 & -1 \\ 3 & -5 \end{vmatrix}\) = ( -15 + 3 ) = -12
(ii) Let \(\begin{bmatrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{bmatrix}\)
By expanding along the first row, we have:
|A| = \(3 \begin{vmatrix} 1 & -2 \\ 3 & 1 \end{vmatrix} + 4 \begin{vmatrix} 1 & -2 \\ 2 & 1 \end{vmatrix} + 5 \begin{vmatrix} 1 & 1 \\ 2 & 3 \end{vmatrix}\)
= 3 ( 1 + 6 ) + 4 ( 1 + 4 ) + 5 ( 3 – 2 )
= 3 ( 7 ) + 4 ( 5 ) + 5 ( 1 )
= 21 + 20 + 5 = 46
(iii) Let A = \(\begin{bmatrix} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{bmatrix}\)
By expanding along the first row, we have:
|A| = \(0 \begin{vmatrix} 0 & -3 \\ 3 & 0 \end{vmatrix} – 1 \begin{vmatrix} -1 & -3 \\ -2 & 0 \end{vmatrix} + 2 \begin{vmatrix} -1 & 0 \\ -2 & 3 \end{vmatrix}\)
= 0 – 1( 0 – 6 ) + 2 ( -3 – 0 )
= – 1 ( -6 ) + 2 ( -3 )
= 6 – 6 = 0
(iv) Let A = \(\begin{bmatrix} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{bmatrix}\)
By expanding along the first column, we have:
|A| = \(2 \begin{vmatrix} 2 & -1 \\ -5 & 0 \end{vmatrix} – 0 \begin{vmatrix} -1 & -2 \\ -5 & 0 \end{vmatrix} + 3 \begin{vmatrix} -1 & -2 \\ 2 & -1 \end{vmatrix}\)
= 2 ( 0 – 5 ) – 0 + 3 ( 1 + 4 )
= -10 + 15 = 5
Q6: If A = \(\begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix}\) , find |A|.
Ans:
Let A = \(\begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix}\)
By expanding along the first row, we have:
|A| = \(1 \begin{vmatrix} 1 & -3 \\ 4 & -9 \end{vmatrix} – 1 \begin{vmatrix} 2 & -3 \\ 5 & -9 \end{vmatrix} – 2 \begin{vmatrix} 2 & 1 \\ 5 & 4 \end{vmatrix}\)
= 1 ( -9 + 12 ) – 1 ( -18 + 15 ) – 2 ( 8 – 5 )
= 1 ( 3 ) – 1 ( -3 ) – 2 ( 3 )
= 3 + 3 – 6
= 6 – 6
= 0
Q7: Find values of x, if
(i) \(\begin{vmatrix} 2 & 4 \\ 2 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}\\\)
(ii) \(\begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix} = \begin{vmatrix} x & 3 \\ 2x & 5 \end{vmatrix}\)
Ans:
(i) \(\begin{vmatrix} 2 & 4 \\ 2 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}\)
\(\boldsymbol{\Rightarrow }\) 2 × 1 – 5 × 4 = 2x × x – 6 × 4
\(\boldsymbol{\Rightarrow }\) 2 – 20 = 2x2 – 24
\(\boldsymbol{\Rightarrow }\) 2x2 = 6
\(\boldsymbol{\Rightarrow }\) x2 = 3
\(\boldsymbol{\Rightarrow }\) x = ± \(\sqrt{ 3 }\)
(ii) \(\begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix} = \begin{vmatrix} x & 3 \\ 2x & 5 \end{vmatrix}\)
\(\boldsymbol{\Rightarrow }\) 2 × 5 – 3 × 4 = x × 5 – 3 × 2x
\(\boldsymbol{\Rightarrow }\) 10 -12 = 5x – 6x
\(\boldsymbol{\Rightarrow }\) -2 = -x
\(\boldsymbol{\Rightarrow }\) x = 2
Q8: If \(\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}\) , then x is equal to
(A) 6
(B) ± 6
(C) – 6
(D) 0
Ans:
\(\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}\\\)\(\\\boldsymbol{\Rightarrow }\) x2 – 36 = 36 – 36
\(\boldsymbol{\Rightarrow }\) x2 – 36 = 0
\(\boldsymbol{\Rightarrow }\) x = ± 6
Hence, the correct answer is B
