Singular and Non Singualar Matrices

Q.

If $A$ and $B$ are two symmetric matrices of the same order. Then, the matrix $AB-BA$ is equal to

1. a symmetric matrix

2. a skew-symmetric matrix

3. a null matrix

4. the identity matrix

Q.

The product of $n$ positive numbers is unity. Their sum is

1. a positive integer

2. equal to $n+\frac{1}{n}$

3. divisible by $n$

4. never less than $n$

Q. Let $\beta$ be a real number. Consider the matrix

$A=\left[\begin{array}{ccc}\beta & 0& 1\\ 2& 1& -2\\ 3& 1& -2\end{array}\right]$. If $A^7-(\beta -1)A^6-\beta A^5$ is a singular matrix, then the value of $9\beta$ is .

Q. 34. Show that positive odd integral powers of a skew- symmetric matrix are skew- symmetric and positive even integral power of a skew- symmetric matrix are symmetric.

Q. $\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{a. If A is an indempotent matrix and I is an}& \\ \text{identity matrix of the same order, then the}& \\ \text{value of}n,\text{such that}(A+I{)}^n=I+127A\text{is}& p.9\\ \text{b. If}(I-A{)}^{-1}=I+A+A^2+....+A^7,& \\ \text{then}{A}^n=0\text{where}n\text{is}& \text{q.}10\\ \text{c. If A is matrix such that}{a}_{ij}=(i+j)(i-j),& \\ \text{then A is singular if order of marix is}& r.7\\ \text{d. If a non-singular matrix A is symmetric,}& \\ \text{show that}{A}^{-1}\text{is also symmetric, then order}& \\ \text{of A can be}& s.8\end{array}$
Which of the following options is/are correct?

1. $a\to r;b\to s;c\to ,r;d\to p$
2. $a\to r;b\to p,q,s;c\to ,p,r;d\to p,q,r,s.$
3. $a\to r;b\to p,,s;c\to ,p,r;d\to r,s.$
4. $a\to s;b\to p,q,s;c\to ,p,r;d\to q,r,s.$

Q. If $A$ and $B$ are different matrices satisfying $A^3=B^3\text{and}{A}^{2}B=B^{2}A,$ then

1. ${\text{det(A}}^2+{\text{B}}^2)\text{must be zero}$
2. $\text{det (A-B) must be zero}$
3. Both ${\text{det(A}}^2+{\text{B}}^2)$ &$\text{det(A-B) must be zero}$
4. ${\text{At least one of det (A}}^2+{\text{B}}^2)\text{or det (A-B) must be zero}$

Q.

If $A=\left[\begin{array}{cc}3& x-1\\ 2x+3& x+2\end{array}\right]$ is a symmetric matrix, then the value of $x$ is

1. $4$

2. $3$

3. $-4$

4. $-3$

Q.

Let $P\left(x\right)=x^2+bx+c$ be a quadratic polynomial with real coefficients such that ${\int }_0^{1}P\left(x\right)dx=1$ and $P\left(x\right)$ leaves remainder $5$ when it is divided by $(x–2)$. Then the value of $9(b+c)$ is equal to :

1. $7$

2. $11$

3. $15$

4. $9$

Q. If A is a skew-symmetric matrix of order 3, then the matrix $A^4$ is

1. skew symmetric
2. diagonal
3. symmetric
4. none of those

Q.

If the data given to construct a triangle ABC is a=5, b=7, $sinA=\frac{3}{4}$ then it is possible to construct

1. Only one triangle

2. two triangle

3. infinitely many triangles

4. No triangle

Q.

The function $f$ is defined by $f\left(x\right)=\left(x+2\right)e^{-x}$ is

1. Decreasing for all $x$.

2. Decreasing in $\left(-\infty ,-1\right)$ and increasing in $\left(-1,\infty \right)$.

3. Increasing for all $x$.

4. Decreasing in $\left(-1,\infty \right)$ and increasing in $\left(-\infty ,-1\right)$.

Q.

The inverse of a symmetric matrix is

Q.

If $f\left(x\right)=e^x$ and $g\left(x\right)=\log e^x$, then which of the following is correct?

1. $f\left(g\left(x\right)\right)\ne g\left(f\left(x\right)\right)$

2. $f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)$

3. $f\left(g\left(x\right)\right)+g\left(f\left(x\right)\right)=0$

4. $f\left(g\left(x\right)\right)-g\left(f\left(x\right)\right)=1$

Q.

What is a singular matrix?

Q. If A and B are two non-zero square matrices of the same order then AB = O implies that both A and B must be singular.

Q. Let $P$ and $Q$ be two matrices different from identity matrix $I$ and null matrix $O$ such that $PQ=QP$ and if $P^6-Q^6=P^5-Q^5=P^4-Q^4=I$ then,

1. $I-P$ is singular
2. $I-Q$ is singular
3. $P+Q=PQ$
4. $(I-P)(I-Q)$ is non singular

Q.

How to find the eigenvectors of a matrix?

Q.

If $A=\{1,2\},B=\{1,3\},$ then $(A\times B)\cup (B\times A)$ is equal to

1. $\left\{\right(1,3),(2,3),(3,1),(3,2),(1,1),(2,1),(1,2\left)\right\}$

2. $\left\{\right(1,3),(3,1),(3,2),(2,3\left)\right\}$

3. $\left\{\right(1,3),(2,3),(3,1),(3,2),(1,1\left)\right\}$

4. None of these

Q.

If A, B are square matrices of order 3, A is non-singular and AB = O, then B is a
(a) null matrix
(b) singular matrix
(c) unit-matrix
(d) non-singular matrix

Q.

Find the adjoint of given matrix.

$\left[\begin{array}{ccc}1& -1& 2\\ 2& 3& 5\\ -2& 0& 1\end{array}\right]$

Q.

Can our answer vary while calculating inverse of a matrix

Q. If the matrix $A=\left[\begin{array}{ccc}8& -6& 2\\ -6& 7& -4\\ 2& -4& \lambda \end{array}\right]$ is singular, then $\lambda =$

1. $3$
2. $4$
3. $2$
4. $5$

Q. If $x,y,z$ are three real numbers and $A=\left[\begin{array}{ccc}1& \cos (x-y)& \cos (x-z)\\ \cos (y-x)& 1& \cos (y-z)\\ \cos (z-x)& \cos (z-y)& 1\end{array}\right]$
then
$A$ is

1. symmetric matrix
2. non-singular matrix
3. not invertibe matrix
4. orthogonal matrix

Q. If
$A=\left[\begin{array}{cccc}{e}^t& e^{-t}\cos t& e^{-t}\sin t& \\ e^t& -e^{-t}\cos t-e^{-t}\sin t& -e^{-t}\sin t+e^{-t}\cos t& \\ e^t& 2e^{-t}\sin t& -2e^{-t}\cos t& \end{array}\right]$,

then $A$ is :

1. invertible only if $t=\pi$.
2. invertible only if $t=\frac{\pi }{2}$.
3. not invertible for any $t\in R$.
4. invertible for all $t\in R$.

Q. Let $A$ be a symmetric matrix such that $A^4=0$ and $B=I+A+A^2+A^3,$ then $B$ is

1. singular matrix
2. symmetric matrix
3. non-singular matrix
4. skew symmetric matrix

Q.

If $f\left(x\right)=Ax+B$ and $f\left(0\right)=f\left(x\right)=2$ then $f\left(1\right)=$

1. $4$

2. $2$

3. $1$

4. $-4$

Q. Let $A$ be a square matrix. If the value of $det\left(A\right)$ is $3$, then the value of $det\left(A^T\right)$ is equal to

1. $3$
2. $0$
3. can't be determined
4. $4$

Q. For what value of $x$ the given matrix $A=\left[\begin{array}{cc}3-2x& x+1\\ 2& 4\end{array}\right]$ is singular matrix.

Q. For what value of $x$, the matrix $A$ is singular?$A=\left[\begin{array}{ccc}3-x& 2& 2\\ 2& 4-x& 1\\ -2& -4& -1-x\end{array}\right]$

1. $x=0,2$
2. $x=1,2$
3. $x=2,3$
4. $x=0,3$

Q. If A is a singular matrix, then write the value of |A|.