Matrices & Determinants · NDA previous year questions

The system of linear equations x + 2y + z = 4, 2x + 4y + 2z = 8 and 3x + 6y + 3z = 10 has

1. A.a unique solution
2. B.infinite many solutions
3. C.no solution
4. D.exactly three solutions

Consider the determinant $$\Delta =\left|\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right|$$ If ${\mathrm{a}}_{13}=yz$, ${\mathrm{a}}_{23}=zx$, ${\mathrm{a}}_{33}=xy$ and the minors of ${\mathrm{a}}_{13},{\mathrm{a}}_{23},{\mathrm{a}}_{33}$ are respectively $(z-y)$, $(z-x)$, $(y-x)$ then what is the value of $\Delta$ ?

1. A.$(z-y)(z-x)(y-x)$
2. B.$(x-y)(y-z)(x-z)$
3. C.$(x-y)(z-x)(y-z)(x+y+z)$
4. D.$(xy+yz+zx)(x+y+z)$

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

If X is a matrix of order 3 × 3, Y is a matrix of order 2 × 3 and Z is a matrix of order 3 × 2, then which of the following are correct? 1. (ZY)X is a square matrix having 9 entries. 2. Y(XZ) is a square matrix having 4 entries. 3. X(YZ) is not defined. Select the correct answer using the code given below :

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

1. A.0
2. B.1
3. C.Δ
4. D.−Δ

1. A.0
2. B.1
3. C.Δ
4. D.−Δ

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

1. A.$\left(\begin{array}{ccc}0& 1/2& 1/2\\ 1/2& 0& 1/2\\ 1/2& 1/2& 0\end{array}\right)$
2. B.$\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)$
3. C.$\cos 2\theta \left(\begin{array}{ccc}0& -1& 1\\ 1& 0& -1\\ -1& 1& 0\end{array}\right)$
4. D.$\cos 2\theta \left(\begin{array}{ccc}0& -1/2& 1/2\\ 1/2& 0& -1/2\\ -1/2& 1/2& 0\end{array}\right)$

1. A.$\left(\begin{array}{ccc}0& 1/2& 1/2\\ 1/2& 0& 1/2\\ 1/2& 1/2& 0\end{array}\right)$
2. B.$\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)$
3. C.$\cos 2\theta \left(\begin{array}{ccc}0& -1& 1\\ 1& 0& -1\\ -1& 1& 0\end{array}\right)$
4. D.$\cos 2\theta \left(\begin{array}{ccc}0& -1/2& 1/2\\ 1/2& 0& -1/2\\ -1/2& 1/2& 0\end{array}\right)$

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

What is the height of the lamp post ?

1. A.6 m
2. B.7 m
3. C.8 m
4. D.9 m

1. A.17 m
2. B.$\frac{50}{3}\mathrm{m}$
3. C.$\frac{40}{3}\mathrm{m}$
4. D.16 m

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

1. A.0
2. B.always positive
3. C.always negative
4. D.positive if p, q, r are positive else negative

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

1. A.0 only
2. B.1 only
3. C.2 only
4. D.Any real value

Let A be a non-singular matrix and B = adj A. Which of the following statements is/are correct? 1. AB = BA 2. AB is a scalar matrix 3. AB can be a null matrix Select the correct answer using the code given below:

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

Consider the following statements in respect of square matrices A and B of same order : 1. If AB is a null matrix, then at least one of A and B is a null matrix. 2. If AB is an identity matrix, then BA = AB. Which of the above statements 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 is the identity matrix of order 3 and B is its transpose, then what is the value of the determinant of the matrix C = A + B?

1. A.1
2. B.2
3. C.4
4. D.8

Let A and B be non-singular matrices of the same order such that AB = A and BA = B. Which of the following statements is/are correct ? 1. $A^2=A$ 2. $A{B}^2=A^{2}B$ Select the correct answer using the code given below:

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

$A=\left[\begin{array}{cc}1& a\\ 0& 1\end{array}\right]$ where $a\in \mathbb{N}$, then is $A^{100}-A^{50}-2A^{25}$ equal to? where I is the identity matrix.

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

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

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

The inverse of a matrix A is given by $\left[\begin{array}{cc}-2& 1\\ \frac{3}{2}& -\frac{1}{2}\end{array}\right]$. What is A equal to?

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

Let $A=\left[\begin{array}{cc}0& 2\\ -2& 0\end{array}\right]$ and $(mI+nA{)}^2=A$ where m, n are positive real numbers and I is the identify matrix. What is (m + n) equal to?

1. A.0
2. B.$\frac{1}{2}$
3. C.1
4. D.$\frac{3}{2}$