Finding Inverse Using Elementary Transformations

Q.

Which of the following would be the Inverse of the matrix A found using Elementary Row Transformations where

A =$\left|\begin{array}{ccc}1& 3& 1\\ 0& 1& 1\\ 3& 6& 1\end{array}\right|$

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

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

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

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

Q. (a)

If $H=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ 3& 1& 2\end{array}\right],G=\left[\begin{array}{ccc}2& 6& -10\\ 0& -4& 8\\ 2& -8& 4\end{array}\right]$

Find $3H-\frac{1}{2}G.$
$\left[3\right]$

(b) If $3A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ x& 2& y\end{array}\right]$ and $A\cdot A^T=I$, then find the value of $x+y$.
$\left[3\right]$

(c) If $A=\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right],B=\left[\begin{array}{cc}4& -2\\ -1& 3\end{array}\right]$ and $I$ is Identity matrix of same order and $A^t$ is the transpose of matrix $A$ find $A^t.B+BI$.
$\left[4\right]$

Q. If $A=\left[\begin{array}{cc}1& -2\\ 3& 0\end{array}\right],B=\left[\begin{array}{cc}-1& 4\\ 2& 3\end{array}\right]$ and $ABC=\left[\begin{array}{cc}4& 8\\ 3& 7\end{array}\right]$, then $C$ equals

1. $\frac{1}{66}\left[\begin{array}{cc}72& -32\\ 57& -29\end{array}\right]$
2. $\frac{1}{66}\left[\begin{array}{cc}-54& -110\\ 3& 11\end{array}\right]$
3. $\frac{1}{66}\left[\begin{array}{cc}-54& 110\\ 3& -11\end{array}\right]$
4. $\frac{1}{66}\left[\begin{array}{cc}-72& 32\\ -57& 29\end{array}\right]$

Q. If $A= $$\left[\begin{array}{cc}0& 1\\ 2& 4\end{array}\right]$$ B= $$\left[\begin{array}{cc}-1& 1\\ 2& 2\end{array}\right]$$ $,

1. $\left[\begin{array}{cc}-4& 5\\ 9& 14\end{array}\right]$
2. $\left[\begin{array}{cc}4& 3\\ 9& 10\end{array}\right]$
3. $\left[\begin{array}{cc}4& -5\\ 9& 14\end{array}\right]$
4. $\left[\begin{array}{cc}-4& 5\\ 14& 9\end{array}\right]$

Q. If $A=$ $\left[\begin{array}{cc}2& 1\\ 0& -2\end{array}\right],$ $B=$ $\left[\begin{array}{cc}4& 1\\ -3& -2\end{array}\right]$ and $C=$ $\left[\begin{array}{cc}-3& 2\\ -1& 4\end{array}\right],$ then $A^2+AC-5B$ =$\left[\begin{array}{cc}17& 13\\ -12& -14\end{array}\right]$

1. True
2. False

Q.

Elementary row transformations can be used to find Inverse for any square matrix.

1. True

2. False

Q. The matrix $a=\left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]$ is

1. Idempotent only
2. Involutary only
3. Involutary and Orthogonal
4. Involutary and Idempotent

Q. If $A=\left(\begin{array}{ccc}8& 5& 2\\ 1& -3& 4\end{array}\right)$ then find $A^T$ and $(A^T{)}^T$

Q. $\underset{n\to \infty }{lim}\left|\begin{array}{c}\frac{1}{1.3}+\frac{1}{3.5}+........+tonterms\end{array}\right|$ is equal to

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

Q. If $A=\left[\begin{array}{cc}2& 3\\ 5& -2\end{array}\right]$ , write $A^{-1}$

Q. If A =$\left[\begin{array}{c}i\\ 0\end{array}\begin{array}{c}0\\ -1\end{array}\right]$, than check whether: ${\text{A}}^2=-\text{I, (}{\text{i}}^2=-1)$

1. True
2. False