Finding Inverse Using Elementary Transformations

Q. Let A be a matrix of order $2\times 2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the sum of all the entries of $A$ is a prime number $p,2<p<8$, then the number of such matrices $A$ is .

Q.

Find out the wrong term: $15,16,22,29,45,70.$

Q. If for any $2\times 2$ square matrix $\text{A, A (adj A)}=\left[\begin{array}{cc}8& 0\\ 0& 8\end{array}\right]$, then find the value of $|A|$.

Q.

How many different rectangles can be drawn on an 8*8 chess board?

Q.

Using appropriate properties, find:

i) $-\frac{2}{3}\times \frac{3}{5}+\frac{5}{2}-\frac{3}{5}\times \frac{1}{6}$

1. $2;\frac{-11}{28}$
2. $3;\frac{-12}{29}$
3. $2;\frac{3}{10}$
4. $4;-3$

Q. If the matrices A, B and A+B are invertible then [A(A+B)^-1 B]^-1 is equal to

Q.

If $3\sin A+5\cos A=5$, then the value of ${(3\cos A-5\sin A)}^2$ is

1. $4$

2. $5$

3. $2$

4. $9$

Q. If $\Delta =\left|\begin{array}{ccc}1& 2cos\theta & 1\\ sin\theta & 1& 3cos\theta \\ 1& sin\theta & 1\end{array}\right|$, then maximum value of $\Delta$ is

1. 1
2. 9
3. 16
4. none of these

Q. If $\Delta =\left|\begin{array}{ccc}abc& b^{2}c& c^{2}b\\ abc& c^{2}a& c{a}^2\\ abc& a^{2}b& b^{2}a\end{array}\right|=0$ $(a,b,c\in R$
and are all different and non-zero), then the value of $a+b+c$ is:

1. $1$
2. $0$
3. $3$
4. $-1$

Q. Let A be a non-singular square matrix of order $3\text{and}B=adj\left(adj\right(adj\left(A^{–1}\right)){)}^{–1}\text{and}C=adj(adj\left(adj\right(B^{–1}\left)\right){)}^{–1}$. Then

1. $BC=|A{|}^{24}I$
2. $adj\left(A\right)=\frac{B}{|A{|}^2}$
3. $adj\left(A\right)=\frac{B}{|A{|}^2}$
4. $C=|A{|}^{21}A$

Q. $\left(\begin{array}{c}n\\ n-r\end{array}\right)+\left(\begin{array}{c}n\\ r+1\end{array}\right)$, whenever $0\le r\le n-1$ is equal to

1. $\left(\begin{array}{c}n\\ r-1\end{array}\right)$
2. $\left(\begin{array}{c}n\\ r\end{array}\right)$
3. $\left(\begin{array}{c}n\\ r+1\end{array}\right)$
4. $\left(\begin{array}{c}n+1\\ r+1\end{array}\right)$

Q.

If $\left|\begin{array}{ccc}x+y+2z& x& y\\ z& y+z+2x& y\\ z& x& z+x+2y\end{array}\right|=k{\left(x+y+z\right)}^3$, then the value of $k$ is

1. $1$

2. $2$

3. $4$

4. $8$

Q.

What is the sum of a matrix $A$ and its negative matrix?

Q.

Solve the following equations using matrix inversion method: $2x-3y+6=0$ and $6x+y+8=0$.

Q.

Physically transpose of a matrix can be thought of as the mirror image of the original matrix. Similarly what is the physical meaning of the inverse of a matrix?

Q. The inverse of $A=\left[\begin{array}{ccc}3& 0& 2\\ 2& 0& -2\\ 0& 1& 1\end{array}\right]$ is

1. $\left[\begin{array}{ccc}\frac{1}{5}& \frac{1}{5}& 0\\ -\frac{1}{5}& \frac{3}{10}& 1\\ \frac{1}{5}& -\frac{3}{10}& 0\end{array}\right]$
2. $\left[\begin{array}{ccc}\frac{-1}{5}& \frac{1}{5}& 1\\ -\frac{1}{5}& \frac{3}{10}& 1\\ \frac{1}{5}& -\frac{3}{10}& 0\end{array}\right]$
3. $\left[\begin{array}{ccc}\frac{2}{5}& \frac{3}{5}& -1\\ -\frac{1}{5}& \frac{3}{5}& 1\\ \frac{1}{5}& -\frac{3}{10}& 0\end{array}\right]$
4. $\left[\begin{array}{ccc}\frac{-1}{5}& \frac{2}{5}& 0\\ -\frac{-3}{5}& \frac{3}{10}& 1\\ \frac{1}{5}& -\frac{3}{10}& 0\end{array}\right]$

Q. Find the number of all possible symmetric matrices of order 3*3 with each entry 1 or2 and whose sum of diagonal elements is equal to 5 is

Q. Find the inverse of the following of the matrices , if it exists.

Q. For a fixed positive integer n, if
$D=\left|\begin{array}{ccc}n!& (n+1)!& (n+2)!\\ (n+1)!& (n+2)!& (n+3)!\\ (n+2)!& (n+3)!& (n+4)!\end{array}\right|$, then $[\frac{D}{(n!{)}^3}-4]$ is___

1. divisible by n.
2. divisible by n+1
3. divisible by n+2
4. divisible by n+3

Q. If $A=\left[\begin{array}{cc}5& 2\\ 2& 1\end{array}\right]$ and matrix $B$ is such that $A{B}^{T}A=B$ and $B^{T}AB=I$ where $I$ is unit matrix of order $2$, then matrix $B^2$ is

1. $\left[\begin{array}{cc}29& 12\\ 12& 5\end{array}\right]$
2. $\left[\begin{array}{cc}1& -2\\ -2& 5\end{array}\right]$
3. $\left[\begin{array}{cc}-1& 2\\ 2& -5\end{array}\right]$
4. $\left[\begin{array}{cc}5& 2\\ 2& 1\end{array}\right]$

Q. $A,B,C$ are three square matrices of order $n$ such that $A=B+C$ and $C$ is a nilpotent matrix with index $2$. If $B$ and $C$ commute, then $A^{10}$ is equal to

1. $B^{10}+12B^{9}C$
2. $B^{10}+10B^{9}C$
3. $B^{10}+12B^{10}C$
4. $B^{10}+10B^{10}C$

Q. Is this possible $a\ne 0.\left|\begin{array}{ccc}x+1& x& x\\ x& x+a& x\\ x& x& x+a^2\end{array}\right|=0$ represents a straight line parallel to the y-axis.

1. True
2. False

Q.

Using elementary transformations, find the inverse of the followng matrix.

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

Q. Using elementary transformation, find the inverse of the matrix $\left[\begin{array}{ccc}2& -3& 3\\ 2& 2& 3\\ 3& -2& 2\end{array}\right]$.

Q. Inverse of matrix by elementary operations

Q.

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

1. True

2. False

Q. For a matrix $A\left(\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right)$, if $U_1,U_2$ and $U_3$ are $3\times 1$ column matrices satisfying $A{U}_1=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),A{U}_2\left(\begin{array}{c}2\\ 3\\ 0\end{array}\right),A{U}_3=\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$ and $U$ is $3\times 3$ matrix whose columns are $U_1,U_2$ and $U_3$
Then sum of the elements of $U^{-1}$ is

1. $0\left(zero\right)$
2. $6$
3. $1$
4. $2/3$

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

1. $\left[\begin{array}{cc}8& 20\\ 7& 9\end{array}\right]$
2. $\left[\begin{array}{cc}8& -20\\ 7& -9\end{array}\right]$
3. $\left[\begin{array}{cc}-8& 20\\ -7& 9\end{array}\right]$
4. $\left[\begin{array}{cc}8& 7\\ -20& -9\end{array}\right]$

Q. If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ then the matrix A is equal to

1. $\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$
2. $\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$
3. $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
4. $\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]$

Q. Using properties of determinants, prove the following:
$\left|\begin{array}{ccc}a& b& c\\ a-b& b-c& c-a\\ b+c& c+a& a+b\end{array}\right|=a^3+b^3+c^3-3abc$