Matrices & Determinants · NDA previous year questions

What is the value of the determinant $\left|\begin{array}{ccc}x+1& x+2& x+4\\ x+3& x+5& x+8\\ x+7& x+10& x+14\end{array}\right|$?

1. A.$x+2$
2. B.$x^2+2$
3. C.2
4. D.$-2$

Consider the following statements: 1. The inverse of a square matrix, if it exists, is unique. 2. If $A$ and $B$ are singular matrices of order $n$, then $AB$ is also a singular matrix of order $n$. Which of the statements given above is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

If the matrix $A=\left(\begin{array}{ccc}2-x& 1& 1\\ 1& 3-x& 0\\ -1& -3& -x\end{array}\right)$ is singular, then what is the solution set $S$?

1. A.$S=\{0,2,3\}$
2. B.$S=\{-1,2,3\}$
3. C.$S=\{1,2,3\}$
4. D.$S=\{2,3\}$

If $\left[\begin{array}{ccc}1& -3& 2\\ 2& -8& 5\\ 4& 2& \lambda \end{array}\right]$ is not an invertible matrix, then what is the value of $\lambda$?

1. A.-1
2. B.0
3. C.1
4. D.2

If $|A|=8$, where $A$ is a square matrix of order 3, then what is $|\mathrm{adj}A|$ equal to?

1. A.16
2. B.24
3. C.64
4. D.512

Consider the following statements in respect of a square matrix $A$ and its transpose $A^T$ : 1. $A+A^T$ is always symmetric. 2. $A-A^T$ is always anti-symmetric. Which of the statements given above is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

If a matrix $A$ is such that $3A^3+2A^2+5A+I=0$, then what is $A^{-1}$ equal to?

1. A.$-(3A^2+2A+5)$
2. B.$3A^2+2A+5I$
3. C.$3A^2-2A-5I$
4. D.$-(3A^2+2A+5I)$

Let $A$ and $B$ be matrices of order $3\times 3$. If $AB=0$, then which of the following can be concluded?

1. A.$A=0$ and $B=0$
2. B.$|A|=0$ and $|B|=0$
3. C.Either $|A|=0$ or $|B|=0$
4. D.Either $A=0$ or $B=0$

If $A$ is a square matrix, then what is $\mathrm{adj}{A}^T-(\mathrm{adj}A{)}^T$ equal to?

1. A.$2|A|$
2. B.$2|A|I$
3. C.Null matrix
4. D.Unit matrix

Which one of the following is correct?

1. A.$\sin 1^{\circ }>\sin 1$
2. B.$\sin 1^{\circ }<\sin 1$
3. C.$\sin 1^{\circ }=\sin 1$
4. D.$\sin 1^{\circ }=\frac{\pi }{180}\sin 1$

1. A.10
2. B.20
3. C.40
4. D.60

1. A.$\left(\begin{array}{ccc}5& 1& 4\\ 2& 6& 3\end{array}\right)$
2. B.$\left(\begin{array}{ccc}2& 6& 3\\ 5& 1& 4\end{array}\right)$
3. C.$\left(\begin{array}{cc}5& 2\\ 1& 6\end{array}\right)$
4. D.$\left(\begin{array}{cc}2& 5\\ 6& 1\end{array}\right)$

Consider the following statements: 1. If $A^{\prime }=A$, then A is a singular matrix, where $A^{\prime }$ is the transpose of A. 2. If A is a square matrix such that $A^3=I$, then A is non-singular. Which of the statements given above is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

Let $A$ be an $n\times n$ matrix. If $\det (\lambda A)={\lambda }^s\det (A)$, what is the value of $s$?

1. A.0
2. B.1
3. C.$-1$
4. D.$n$

If $A$ be a real skew-symmetric matrix of order $n$ such that $A^2+I=0$, $I$ being the identity matrix of the same order as that of $A$, then what is the order of $A$?

1. A.3
2. B.Odd
3. C.Prime number
4. D.Even

Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=[a_{ij}]$, where i, j = 1, 2. If its inverse matrix is $[b_{ij}]$, what is $b_{22}$?

1. A.-2
2. B.1
3. C.$\frac{3}{2}$
4. D.$-\frac{1}{2}$

The angle A lies in the third quadrant and it satisfies the equation $4({\sin }^{2}x+\cos x)=1$. What is the measure of the angle A?

1. A.$225^{\circ }$
2. B.$240^{\circ }$
3. C.$210^{\circ }$
4. D.None of the above

1. A.$\left[\begin{array}{cc}1& 3\\ -2& 1\end{array}\right]$
2. B.$\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]$
3. C.$\left[\begin{array}{cc}3& 2\\ -1& 5\end{array}\right]$
4. D.$\left[\begin{array}{cc}3& 2\\ 1& 5\end{array}\right]$

1. A.2
2. B.1
3. C.$-1$
4. D.0

If a matrix $A$ is symmetric as well as anti-symmetric, then which one of the following is correct ?

1. A.$A$ is a diagonal matrix.
2. B.$A$ is a null matrix.
3. C.$A$ is a unit matrix.
4. D.$A$ is a triangular matrix.

1. A.$A$ is symmetric matrix.
2. B.$A$ is anti-symmetric matrix.
3. C.$A$ is a singular matrix.
4. D.$A$ is a non-singular matrix.

1. A.12
2. B.$-12$
3. C.7
4. D.$-7$

1. A.$A$
2. B.$-A$
3. C.Null matrix
4. D.Identity matrix

A matrix $X$ has $(a+b)$ rows and $(a+2)$ columns; and a matrix $Y$ has $(b+1)$ rows and $(a+3)$ columns. If both $XY$ and $YX$ exist, then what are the values of $a,b$ respectively ?

1. A.3, 2
2. B.2, 3
3. C.2, 4
4. D.4, 3

In a football championship 153 matches were played. Every team played one match with each other team. How many teams participated in the championship ?

1. A.21
2. B.18
3. C.17
4. D.15