Scalar Multiplication of a Matrix

Q. If $A$ is a skew-symmetric matrix of order $3$, then prove that $detA=0$.

Q. Let $A=\left[\begin{array}{ccc}-3& 0& 2\\ 1& x& 5\\ -2& 0& x^2\end{array}\right],B=\left[\begin{array}{c}2\\ b\\ -1\end{array}\right]$ and $C=\left[\begin{array}{ccc}3& 5& 1\end{array}\right].$ If $\text{tr}$ denotes the trace of a matrix, then the number of integral values of $b$ for which $\text{tr}\left(ABC\right)\le -18$ for all $x\in R,$ is

1. $3$
2. $4$
3. $5$
4. $6$

Q.

Show that the Signum function $f:R\to R,$ given by $f\left(x\right)=\{\begin{array}{c}1,ifx>0\\ 0,ifx=0\\ -1,ifx<0\end{array}$ is neither one-one nor onto.

Q. Let A be a non singular, symmetric matrix of order three such that $A=adj(A+A^T)$, then

1. $adj\left(A^{-1}\right)=64A$
2. $A^{-1}=\frac{A}{16}$
3. $A^{-1}=\frac{A}{64}$
4. $adj\left(A^{-1}\right)=\frac{A}{1024}$

Q. If $A=\left[\begin{array}{cc}0& 2\\ 3& -4\end{array}\right]$ and $kA=\left[\begin{array}{cc}0& 3a\\ 2b& 24\end{array}\right]$, then the values of k, a, b are respectively.

1. -6, -12, -18
2. -6, 4, 9
3. -6, -4, -9
4. -6, 12, 18

Q. If $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, find $AB$ and $BA.$ Prove that $AB\ne BA$.

Q.

Show that the function defined by g(x)=x-[x] is discontinuous at all integral points. Here, [x] denotes the greatest integer less than or equal to x.

Q. If $A=\left[\begin{array}{cc}2& -1\\ 3& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 4\\ 7& 2\end{array}\right]$, then the value $3A-2B$ is

1. $\left[\begin{array}{cc}3& -11\\ -4& -1\end{array}\right]$
2. $\left[\begin{array}{cc}4& -1\\ -5& -1\end{array}\right]$
3. $\left[\begin{array}{cc}4& -11\\ -5& -1\end{array}\right]$
4. $\left[\begin{array}{cc}4& -11\\ -3& -1\end{array}\right]$

Q. Simplify

Q. The value of deteminant $\left|\begin{array}{ccc}a+b& a& b\\ a& a+c& c\\ b& c& b+c\end{array}\right|$ is equal to

1. $4abc$
2. $abc$
3. $a^2{b}^2{c}^2$
4. $4a^{2}bc$

Q.

Let $A=\left[\begin{array}{ccc}1& sin\theta & 1\\ -sin\theta & 1& sin\theta \\ -1& -sin\theta & 1\end{array}\right]$, where $0\le \theta \le 2\pi$, then

a) det A=0
b) det $Aϵ(2,\infty )$
c) det $Aϵ(2,4)$
d) det $A\epsilon [2,4]$

Q.

Find the matrix X so that X $\left[\begin{array}{ccc}1& 2& 3\\ 4& 4& 6\end{array}\right]=\left[\begin{array}{ccc}-7& -8& -9\\ 2& 4& 6\end{array}\right]$.

Q. Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right],a,b\in N.$ Then

1. There does not exist any $B$ such that $AB=BA$
2. There exists more than one but finite number of $B^{\prime }s$ such that $AB=BA$
3. There exists exactly one $B$ such that $AB=BA.$
4. There exists infinitely many $B^{\prime }s$ such that $AB=BA$

Q. If $A=\left[\begin{array}{cc}\cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{array}\right]$ and $A+A^T=I,$ where $I$ is $2\times 2$ unit matrix and $A^T$ is the transpose of $A,$ then the value of $\theta$ is equal to

1. $\frac{\pi }{6}$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{3}$
4. $\frac{3\pi }{2}$

Q. If $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 1& 0\end{array}\right],B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 3& 0\\ 0& 4& 5\end{array}\right],$ then $Tr\left(AB\right)$ is

Q.

Compute the indicated product.
$$\left[\begin{array}{ccc}3& -1& 3\\ -1& 0& 2\end{array}\right]$$$$\left[\begin{array}{cc}2& -3\\ 1& 0\\ 3& 1\end{array}\right]$$

Q.

Compute the indicated product.
$$\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]$$$$\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right]$$

Q.

Simplify :

$-3a-\left\{2a+5b-(a-b)\right\}-6a-(3a-\overline{4a+b})$

Q.

Compute the following:
$\left(iii\right)\left[\begin{array}{ccc}-1& 4& -6\\ 8& 5& 16\\ 2& 8& 5\end{array}\right]+\left[\begin{array}{ccc}12& 7& 6\\ 8& 0& 5\\ 3& 2& 4\end{array}\right]$

Q. If and then compute .

Q. Suppose $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is a real matrix with nonzero entries, $ad–bc=0$, and $A^2=A$. Then $a+d$ equals

1. 1
2. 4
3. 3
4. 2

Q. If a skew-symmetric matrix of order $2\times 2$ is formed with zero and cube roots of unity, then the determinant of such matrices formed can be

1. $0$
2. $1$
3. $\omega$
4. ${\omega }^2$

Q.

If $\left[\begin{array}{cc}xy& 4\\ z+6& x+y\end{array}\right]=\left[\begin{array}{cc}8& w\\ 0& 6\end{array}\right]$ then find the values of x, y, z and w.

Q. If $A=\left[\begin{array}{cc}0& 2\\ 3& -4\end{array}\right]$ and $kA=\left[\begin{array}{cc}0& 3a\\ 2b& 24\end{array}\right]$, then the values of k, a, b are respectively.

1. -6, 12, 18
2. -6, -12, -18
3. -6, 4, 9
4. -6, -4, -9

Q. If $m[-34]+n[4-3]=[10-11],$ then $3m+7n=$

1. $5$
2. $10$
3. $1$
4. $3$

Q. Multiplication of $\left[\begin{array}{ccc}4& 1& 1\\ 2& 3& 1\end{array}\right]$ with $5$ gives

1. $\left[\begin{array}{ccc}20& 1& 1\\ 2& 3& 1\end{array}\right]$
2. $\left[\begin{array}{ccc}4& 1& 1\\ 10& 15& 5\end{array}\right]$
3. None of the above
4. $\left[\begin{array}{ccc}20& 5& 5\\ 10& 15& 5\end{array}\right]$

Q. If $A=(\begin{array}{cc}3& 3\\ 7& 6\end{array}),B=(\begin{array}{cc}8& 7\\ 0& 9\end{array})$ and $C=(\begin{array}{cc}2& -3\\ 4& 6\end{array})$, find (A + B)C and AC + BC.
Is (A + B)C = AC + BC?

Q. Prove that $\left|\begin{array}{ccc}a& a+b& a+b+c\\ 2a& 3a+2b& 4a+3b+2c\\ 3a& 6a+3b& 10a+6b+3c\end{array}\right|=a^3$.

Q. If $A=\left[\begin{array}{cc}3& a\\ -4& 8\end{array}\right],B=\left[\begin{array}{cc}c& 4\\ -3& 0\end{array}\right],C=\left[\begin{array}{cc}-1& 4\\ 3& b\end{array}\right]$ & $3A-2B=6C$, find the values of $a,b$ and $c.$

Q. If $\left[\begin{array}{ccc}X-Y& 2& -2\\ 4& X& 6\end{array}\right]+\left[\begin{array}{ccc}3& -2& 2\\ 1& 0& -1\end{array}\right]=\left[\begin{array}{ccc}6& 0& 0\\ 5& 2X+Y& 5\end{array}\right]$,
then

1. $X=\frac{3}{2}$
2. $X=\frac{5}{2}$
3. $Y=-\frac{3}{2}$
4. $Y=-\frac{5}{2}$