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Maths (commerce) MCQ

Matrices (p-1) · Maharashtra Board STD 12 Commerce / Arts (MSBSHSE - English Medium). 15 questions.

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MCQ

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15 questions · auto-graded multiple-choice test.

MCQ 11 Mark
If $A =\left[\begin{array}{ll}\alpha & 4 \\ 4 & \alpha\end{array}\right]$ and $\left| A ^3\right|=729$ then $\alpha=$
  • A
    \pm 3
  • B
    \pm 4
  • ✓
    \pm 5
  • D
    \pm 6
Answer
Correct option: C.
\pm 5
(C) \pm 5
Hint:
$
\begin{aligned}
& | A |=\left|\begin{array}{ll}
\alpha & 4 \\
4 & \alpha
\end{array}\right|=\alpha^2-16 \\
& \therefore\left| A ^3\right|=| A |^3=\left(\alpha^2-16\right)^3=729 \\
& \therefore \alpha^2-16=9 \\
& \therefore \alpha^2=25 \\
& \therefore \alpha= \pm 5
\end{aligned}
$
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MCQ 21 Mark
If a $3 \times 3$ matrix $B$ has its inverse equal to $B$, then $B^2=$
  • A
    $\left[\begin{array}{lll}0 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{array}\right]$
  • B
    $\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 1\end{array}\right]$
  • C
    $\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{array}\right]$
  • ✓
    $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
Answer
Correct option: D.
$\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
(d) $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
Hint:
$
\begin{aligned}
& B ^{-1}= B \\
& \therefore B ^2= B \cdot B ^{-1}=1
\end{aligned}
$
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MCQ 31 Mark
If $A^2+m A+n l=O$ and $n \neq 0,|A| \neq 0$, then $A^{-1}=$ ________
  • A
    $\frac{-1}{m}(A+n l)$
  • ✓
    $\frac{-1}{n}(A+m l)$
  • C
    $\frac{-1}{m}(I+m A)$
  • D
    $( A + mnl )$
Answer
Correct option: B.
$\frac{-1}{n}(A+m l)$
(b) $\frac{-1}{n}(A+m l)$
Hint:
$
\begin{aligned}
& A ^2+ mA + nl =0 \\
& \therefore\left( A ^2+ mA + nl \right) \cdot A ^{-1}=0 \cdot A ^{-1} \\
& \therefore A \left( AA^{-1 } \right)+ m \left( AA A ^{-1}\right)+ nl A ^{-1}=0 \\
& \therefore Al + ml + nA A ^{-1}=0 \\
& \therefore nA A ^{-1}=- A - ml \\
& \therefore A ^{-1}=\frac{-1}{n}( A + ml )
\end{aligned}
$
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MCQ 41 Mark
If $A =$ diag. $\left[ d _1, d _2, d _3, \ldots, d _{ n }\right]$, where $d _1 \neq 0$, for $i =1,2,3, \ldots \ldots ., n$, then $A ^{-1}=$
  • ✓
    $\operatorname{diag}\left[1 / d _1, 1 / d _2, 1 / d _3, \ldots, 1 / d _{ n }\right]$
  • B
    D
  • C
    1
  • D
    $O$
Answer
Correct option: A.
$\operatorname{diag}\left[1 / d _1, 1 / d _2, 1 / d _3, \ldots, 1 / d _{ n }\right]$
(a) diag $\left[1 / d _1, 1 / d _2, 1 / d _3, \ldots, 1 / d _{ n }\right]$
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MCQ 51 Mark
Adjoint of $\left[\begin{array}{ll}2 & -3 \\ 4 & -6\end{array}\right]$ is
  • ✓
    $\left[\begin{array}{ll}-6 & 3 \\ -4 & 2\end{array}\right]$
  • B
    $\left[\begin{array}{cc}6 & 3 \\ -4 & 2\end{array}\right]$
  • C
    $\left[\begin{array}{cc}-6 & -3 \\ 4 & 2\end{array}\right]$
  • D
    $\left[\begin{array}{cc}-6 & 3 \\ 4 & -2\end{array}\right]$
Answer
Correct option: A.
$\left[\begin{array}{ll}-6 & 3 \\ -4 & 2\end{array}\right]$
(a) $\left[\begin{array}{ll}-6 & 3 \\ -4 & 2\end{array}\right]$
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MCQ 61 Mark
If $A =\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right]$, then $|\operatorname{adj} A |=$
  • A
    $a^{12}$
  • B
    $a^9$
  • ✓
    $a^6$
  • D
    $a ^{-3}$
Answer
Correct option: C.
$a^6$
(c) $a 6$
Hint:
$
\begin{aligned}
& \operatorname{adj} A =\left[\begin{array}{ccc}
a^2 & 0 & 0 \\
0 & a^2 & 0 \\
0 & 0 & a^2
\end{array}\right] \\
& \therefore|\operatorname{adj} A |= a ^2\left( a ^4-0\right)= a ^6
\end{aligned}
$
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MCQ 71 Mark
The matrix $\left[\begin{array}{ccc}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$ is
  • A
    identity matrix
  • B
    diagonal matrix
  • C
    scalar matix
  • ✓
    null matrix
Answer
Correct option: D.
null matrix
(d) null matrix
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MCQ 81 Mark
The matrix $\left[\begin{array}{ccc}8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8\end{array}\right]$ is
  • A
    identity matrix
  • ✓
    scalar matrix
  • C
    null matrix
  • D
    diagonal matrix
Answer
Correct option: B.
scalar matrix
(b) scalar matrix
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MCQ 91 Mark
If $x+y+z=3, x+2 y+3 z=4, x+4 y+9 z=6$, then $(y, z)=$________
  • A
    $(-1,0)$
  • ✓
    $(1,0)$
  • C
    $(1,-1)$
  • D
    $(-1,1)$
Answer
Correct option: B.
$(1,0)$
(b) $(1,0)$
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MCQ 101 Mark
If $A =\left[\begin{array}{cc}1 & 2 \\ -3 & -1\end{array}\right], B =\left[\begin{array}{rr}-1 & 0 \\ 1 & 5\end{array}\right]$ then $AB =$
  • A
    $\left[\begin{array}{rr}1 & -10 \\ 1 & 20\end{array}\right]$
  • B
    $\left[\begin{array}{rr}1 & 10 \\ -1 & 20\end{array}\right]$
  • ✓
    $\left[\begin{array}{ll}1 & 10 \\ 1 & 20\end{array}\right]$
  • D
    $\left[\begin{array}{rr}1 & 10 \\ -1 & -20\end{array}\right]$
Answer
Correct option: C.
$\left[\begin{array}{ll}1 & 10 \\ 1 & 20\end{array}\right]$
(c) $\left[\begin{array}{ll}1 & 10 \\ 1 & 20\end{array}\right]$
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MCQ 111 Mark
If $A$ is a non-singular matrix, then $\operatorname{det}\left( A ^{-1}\right)=$
  • A
    1
  • B
    $0$
  • C
    $\operatorname{det}( A )$
  • ✓
    $1 / \operatorname{det}(A)$
Answer
Correct option: D.
$1 / \operatorname{det}(A)$
(d) $1 / \operatorname{det}(A)$
Hint:
$
\begin{aligned}
& A A^{-1}=1 \\
& \therefore|A| \cdot\left|A^{-1}\right|=1 \\
& \therefore\left|A^{-1}\right|=\frac{1}{|A|}
\end{aligned}
$
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MCQ 121 Mark
If $A$ is a $2 \times 2$ matrix such that $A(\operatorname{adj} A)=\left[\begin{array}{ll}5 & 0 \\ 0 & 5\end{array}\right]$, then $|A|=$
  • A
    $0$
  • ✓
    5
  • C
    10
  • D
    25
Answer
Correct option: B.
5
(b) 5
Hint:
$A(\operatorname{adj} A)=|A| . \mid$
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MCQ 131 Mark
If $A=\left[\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right]$ then $A^{-1}=$
  • A
    $\left[\begin{array}{rr}3 & -5 \\ 1 & 2\end{array}\right]$
  • ✓
    $\left[\begin{array}{rr}3 & -5 \\ -1 & 2\end{array}\right]$
  • C
    $\left[\begin{array}{rr}3 & 5 \\ -1 & 2\end{array}\right]$
  • D
    $\left[\begin{array}{ll}3 & -5 \\ 1 & -2\end{array}\right]$
Answer
Correct option: B.
$\left[\begin{array}{rr}3 & -5 \\ -1 & 2\end{array}\right]$
(b) $\left[\begin{array}{rr}3 & -5 \\ -1 & 2\end{array}\right]$
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MCQ 141 Mark
If $A$ and $B$ square matrices of order $n \times n$ such that $A^2-B^2=(A-B)$ $(A+B)$, then which of the following will be always true?
  • ✓
    $AB = BA$
  • B
    either $A$ or $B$ is a zero matrix
  • C
    either of $A$ and $B$ is an identity matrix
  • D
    $A = B$
Answer
Correct option: A.
$AB = BA$
(a) $AB = BA$
Hint:
$
\begin{aligned}
& A^2-B^2=(A-B)(A+B) \\
& \therefore A^2-B^2=A^2+A B-B A-B^2 \\
& \therefore 0=A B-B A \\
& \therefore A B=B A
\end{aligned}
$
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MCQ 151 Mark
If $AX = B$, where $A = \begin{bmatrix} -1 & 2 \\ 2 & -1\end{bmatrix} , B =\begin{bmatrix} 1 \\ 1\end{bmatrix} $ then $X =$
  • A
    $ \begin{bmatrix} \frac{3}{5} \\ \frac{3}{7}\end{bmatrix}$
  • B
    $ \begin{bmatrix} \frac{7}{3} \\ \frac{5}{3}\end{bmatrix} $
  • ✓
    $ \begin{bmatrix} 1 \\ 1\end{bmatrix} $
  • D
    $ \begin{bmatrix} 1 \\ 2\end{bmatrix} $
Answer
Correct option: C.
$ \begin{bmatrix} 1 \\ 1\end{bmatrix} $
(c) $ \begin{bmatrix} 1 \\ 1\end{bmatrix} $
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