Matrices And Determinants

If $A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & - 1 \end{matrix}]$ , then $A^{2}$ is equal to:
1. A null matrix
2. A unit matrix
3. -A
4. A
If AB = A and BA = B, where A nad B are square matrices, then:
1. $B^{2} = B a n d A^{2} = A$
2. $B^{2} \neq B a n d A^{2} = A$
3. $B^{2} = B a n d A^{2} \neq A$
4. $B^{2} \neq B a n d A^{2} \neq A$
Which of the following is not correct?
1. $|\begin{matrix} A \end{matrix}| = |\begin{matrix} A^{T} \end{matrix}|, w h e r e A = {[\begin{matrix} a_{i j} \end{matrix}]}_{3 \times 3}$
2. $|k \begin{matrix} A \end{matrix}| = k^{3} |\begin{matrix} A \end{matrix}|, w h e r e A = {[\begin{matrix} a_{i j} \end{matrix}]}_{3 \times 3}$
3. If A is a skew-symmetric matrix of odd order, then $|\begin{matrix} A \end{matrix}| = 0$
4. $|\begin{matrix} a + b & c + d \\ e + f & g + h \end{matrix}| = |\begin{matrix} a & c \\ e & g \end{matrix}| + |\begin{matrix} b & d \\ f & h \end{matrix}|$
Which of the following is not correct in a given determinant of A, where ${[\begin{matrix} A_{i j} \end{matrix}]}_{3 \times 3}$ :
1. Order of minor is less than order of the det (A)
2. Minor of an element can never be equal to cofactor of the same element
3. Value of a determinant is obtained by multiflying elements of a row or column by corresponding cofactors
4. Order of minors and cofactors of elements of A is same
If A is an invertible matrix, then which of the following is not true?
1. ${(A^{2})}^{- 1} = {(\begin{matrix} A^{- 1} \end{matrix})}^{2}$
2. $|\begin{matrix} A^{- 1} \end{matrix}| = {|\begin{matrix} A \end{matrix}|}^{- 1}$
3. ${(A^{T})}^{- 1} = {(\begin{matrix} A^{- 1} \end{matrix})}^{T}$
4. $|A| \neq 0$
If $A = |\begin{matrix} 3 & 4 \\ 2 & 4 \end{matrix}|$ , $B = |\begin{matrix} - 2 & - 2 \\ 0 & - 1 \end{matrix}|$ , then ${(A + B)}^{- 1}$
1. Is a skew-symmetric matrix
2. $A^{- 1} + B^{- 1}$
3. Does not exist
4. None of these
If $S = |\begin{matrix} a & b \\ c & d \end{matrix}|$ , then adj S is:
1. $|\begin{matrix} - d & - b \\ - c & a \end{matrix}|$
2. $|\begin{matrix} d & - b \\ - c & a \end{matrix}|$
3. $|\begin{matrix} d & b \\ c & a \end{matrix}|$
4. $|\begin{matrix} d & c \\ b & a \end{matrix}|$
If $A = |\begin{matrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{matrix}|$ , then the value of $|\begin{matrix} a d j A \end{matrix}|$ is:
1. $a^{27}$
2. $a^{9}$
3. $a^{6}$
4. $a^{2}$
Let $|\begin{matrix} x & 2 & x \\ x^{2} & x & 6 \\ x & x & 6 \end{matrix}| = a x^{4} + b x^{3} + c x^{2} + d x + e$ , Then the value of $5 a + 4 b + 3 c + 2 d + e$ is equal to:
1. 0
2. -16
3. 16
4. None of these
The value of the determinant $|\begin{matrix} a^{2} & a & 1 \\ \cos (n x) & \cos (n + 1) x & \cos (n + 2) x \\ \sin (n x) & \sin (n + 1) x & \sin (n + 2) x \end{matrix}|$ is independent of:
1. n
2. a
3. x
4. None of these
If ${[\begin{matrix} \cos (\frac{2 π}{7}) & - \sin (\frac{2 π}{7}) \\ \sin (\frac{2 π}{7}) & \cos (\frac{2 π}{7}) \end{matrix}]}^{k} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ , then the least positive integral value of k is:
1. 3
2. 4
3. 6
4. 7
If A and B are two matrices such that A + B and AB are both defined, then:
1. A and B are two matrices not necessarily of same order
2. A and B are square matrices of same order
3. Number of columns of A is equal to the number of rows of B
4. A = B
If the matrix AB is zero, then:
1. It is not necessary that either A = O or, B = O
2. A = O or B = O
3. A = O and B = O
4. All the above statements are wrong
If A, B are square matrices of order 3, A is non-singular and AB = O, then B is a:
1. Null matrix
2. Singular matrix
3. Unit matrix
4. Non-singular matrix
If $|\begin{matrix} 2 a & x_{1} & y_{1} \\ 2 b & x_{2} & y_{2} \\ 2 c & x_{3} & y_{3} \end{matrix}| = \frac{a b c}{2} \neq 0$ , then the area of the triangle whose vertices are $(\frac{x_{1}}{a}, \frac{y_{1}}{a})$ , $(\frac{x_{2}}{b}, \frac{y_{2}}{b})$ , $(\frac{x_{3}}{c}, \frac{y_{3}}{c})$ is:
1. $\frac{1}{4}$
2. $\frac{1}{4} a b c$
3. $\frac{1}{8}$
4. $\frac{1}{8} a b c$
The value of the determinant $|\begin{matrix} x & x + y & x + 2 y \\ x + 2 y & x & x + y \\ x + y & x + 2 y & x \end{matrix}|$ is:
1. $9 x^{2} (x + y)$
2. $9 y^{2} (x + y)$
3. $3 y^{2} (x + y)$
4. $7 x^{2} (x + y)$
If B is a non-singular matrix and A is a square matrix, then det $(B^{- 1} A B)$ is equal to:
1. Det $(A^{- 1})$
2. Det $(B^{- 1})$
3. Det (A)
4. Det (B)
If A satisfies the equation $x^{3} - 5 x^{2} + 4 x + λ = 0$ , then $A^{- 1}$ exists if:
1. $λ \neq 1$
2. $λ \neq 2$
3. $λ \neq -1$
4. $λ \neq 0$
The maximum value of $∆ = |\begin{matrix} 1 & 1 & 1 \\ 1 & 1 + \sin (θ) & 1 \\ 1 + \cos (θ) & 1 & 1 \end{matrix}|$ is ( $θ$ is real):
1. $\frac{1}{2}$
2. $\frac{\sqrt{3}}{2}$
3. $\sqrt{2}$
4. $- \frac{\sqrt{3}}{2}$
There are two values of a which makes the determinant $∆ = |\begin{matrix} 1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2 a \end{matrix}|$ equal to 86. The sum of these two values is:
1. 4
2. 5
3. -4
4. 9
If $A = |\begin{matrix} 1 & - 1 \\ 2 & - 1 \end{matrix}|$ , $B = |\begin{matrix} a & 1 \\ b & - 1 \end{matrix}|$ and ${(A + B)}^{2} = A^{2} + B^{2}$ , then values of a and b are:
1. a = 4, b = 1
2. a = 1, b = 4
3. a = 0, b = 4
4. a = 2, b = 4
If A is square matrix such that $A^{2} = A$ , then ${(I + A)}^{3} - 7 A$ is equal to:
1. A
2. I - A
3. I
4. 3A
If A and B are symmetric matrices, then ABA is:
1. Symmetric matrix
2. Skew-symmetric matrix
3. Diagonal matrix
4. Scalar matrix
The number of all possible matrices of order $3 \times 3$ with each entry 0 and 1 is:
1. 27
2. 18
3. 81
4. 512
The value of $|\begin{matrix} 5^{2} & 5^{3} & 5^{4} \\ 5^{3} & 5^{4} & 5^{5} \\ 5^{4} & 5^{5} & 5^{6} \end{matrix}|$ is:
1. $5^{2}$
2. 0
3. $5^{13}$
4. $5^{9}$
If $|\begin{matrix} 2 x & 5 \\ 8 & x \end{matrix}| = |\begin{matrix} 6 & - 2 \\ 7 & 3 \end{matrix}|$ , then the value of x is:
1. 3
2. $\pm 3$
3. $\pm 6$
4. 6
If $A = \frac{1}{3} [\begin{matrix} 1 & 1 & 2 \\ 2 & 1 & - 2 \\ x & 2 & y \end{matrix}]$ satisfies $A^{T} A = I$ , then x + y is:
1. 3
2. 0
3. -3
4. 1
If a matrix A is such that $3 A^{3} + 2 A^{2} + 5 A + I = 0$ , then $A^{- 1}$ is equal to:
1. $- (3 A^{2} + 2 A + 5)$
2. $3 A^{2} + 2 A + 5$
3. $3 A^{2} - 2 A - 5$
4. None of these
The value of the determinant $|\begin{matrix} a - b & b + c & a \\ b - a & c + a & b \\ c - a & a + b & c \end{matrix}|$ is:
1. $a^{3} + b^{3} + c^{3}$
2. $3 b c$
3. $a^{3} + b^{3} + c^{3} - 3 a b c$
4. None of these
If $A = [\begin{matrix} 2 & - 1 & 3 \\ - 4 & 5 & 1 \end{matrix}]$ and $B = [\begin{matrix} 2 & 3 \\ 4 & - 2 \\ 1 & 5 \end{matrix}]$ , then:
1. Only AB is defined
2. Only BA is defined
3. AB and BA both are defined
4. AB and BA both are not defined

Matrices And Determinants

Quiz