If \(\left[ \begin{matrix} x & +3y & y \\ 7 & -x & 4 \end{matrix} \right] \)=\(\begin{bmatrix} 4 & -1 \\ 0 & 4 \end{bmatrix}\), find the values of x and y.

1

(a)

\(\left[ \begin{matrix} x & +3y & y \\ 7 & -x & 4 \end{matrix} \right] =\begin{bmatrix} 4 & -1 \\ 0 & 4 \end{bmatrix}\)
\(\Rightarrow x+3y=4;y=-1;7-x=0\)
\(\Rightarrow x=7,y=-1\)

If matrix A = \([\begin{matrix} 1 & 2 & 3 \end{matrix}]\) write AA' , where A' is the transpose of matrix A.

1

(a)

\(AA'= \left[ \begin{matrix} 1 & 2 & 3 \end{matrix} \right] \left[ \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right] =\left[ 1+4+9 \right] =\left[ 14 \right] \)

If \(\left[ \begin{matrix} y & +2x & 5 \\ & -x & 3 \end{matrix} \right] =\begin{bmatrix} 7 & 5 \\ -2 & 3 \end{bmatrix}\), find the value of y.

1

(a)

\(y+2x=7,-x=-2\Rightarrow x=2,y=3\)

If \(A=\left[ { a }_{ ij } \right] =\left[ \begin{matrix} 2 & 3 & -5 \\ 1 & 4 & 9 \\ 0 & 7 & -2 \end{matrix} \right] andb=\left[ { b }_{ ij } \right] =\left[ \begin{matrix} 2 & 1 & -1 \\ -3 & 4 & 4 \\ 1 & 5 & 2 \end{matrix} \right] ,then\quad find\quad { a }_{ 22 }+{ b }_{ 21 }.\)

1

(a)

\({ a }_{ 22 }+{ b }_{ 21 }\)
=4 - 3 = 1

If \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 3 & 1 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix}\), find the value of k.

1

(a)

\(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 3 & 1 \\ 2 & 5 \end{bmatrix}=\left[ \begin{matrix} 3\quad + & 4 \\ 9\quad + & 8 \end{matrix}\quad \begin{matrix} 1\quad + & 10 \\ 3\quad + & 20 \end{matrix} \right] =\begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix}\)
\(\Rightarrow k=9+8=17\)

If A and B are invertible matrices of the same order, then (AB)^–1 = B^–1 A^–1.

1

(a)

6

Let A = [a_ij]be a matric of order 2 x 3 and a_ij = \(\frac { i-j }{ i+j } \), write the value of a₂₃

1

(a)

-1/5

if \(\left[ \begin{matrix} a+b & 2 \\ 5 & ab \end{matrix} \right] =\left[ \begin{matrix} 6 & 2 \\ 5 & 8 \end{matrix} \right] \)find the relation between a and b 

1

(a)

a = 2b {a=4, b=2}

If, A = |aij| = \(\left[ \begin{matrix} 2 & 3 & -5 \\ 1 & 4 & 9 \\ 0 & 7 & -2 \end{matrix} \right] \)and B =|bij| =\(\left[ \begin{matrix} 2 & -1 \\ -3 & 4 \\ 1 & 2 \end{matrix} \right] \) Write the value of
(i)a₂₂ + b₂₁
(ii) a₁₁b₁₁ +a₂₂b₂₂ 

1

(a)

1, 20

Write the order of matrix B if any matrix of order m x n such that AB and BA both are defined

1

(a)

n x m