  Question:

Find the order of the following matrices.

 A = $~\left[ \begin{matrix} 2 & 3 \\ -5 & 6 \\ \end{matrix} \right]$ B = $~\left[ \begin{matrix} 2 & 0 \\ 3 & 5 \\ \end{matrix} \right]$ C = $\left[ \begin{matrix} 2 & 4 \\ \end{matrix} \right]$ D = $\left[ \begin{matrix} 4 \\ 0 \\ 6 \\ \end{matrix} \right]$ E = $\left[ \begin{matrix} a & d \\ b & e \\ c & f \\ \end{matrix} \right]$ F = $\left[ 2 \right]$ G=$~\left[ \begin{matrix} 2 & 3 & 0 \\ 1 & 2 & 3 \\ 2 & 4 & 5 \\ \end{matrix} \right]$ H = $\left[ \begin{matrix} 2 & 3 & 4 \\ 1 & 0 & 6 \\ \end{matrix} \right]$

Difficulty: Easy

Solution:

Order of the Matrix:

The number of rows and columns in a Matrix specifies its order.

Ans.   (i)      Matrix A has two rows and two columns

So, its order = number of rows x number of columns = 2-by-2.

Ans.   (ii)     Matrix B has two rows and two columns

So, its order = number of rows x number of columns = 2-by-2.

Ans.   (iii)    Matrix C has one row and two columns

So, its order = number of rows x number of columns = 1-by-2.

Ans.   (iv)    Matrix D has three rows and one column

So, its order = number of rows x number of columns = 3-by-1.

Ans.   (v)     Matrix E has three rows and two columns

So, its order = number of rows x number of columns = 3-by-2.

Ans.   (vi)    Matrix F has one row and one column

So, its order = number of rows x number of columns = 1-by-1.

Ans.   (vii)   Matrix G has three rows and three columns

So, its order = number of rows x number of columns = 3-by-3.

Ans.   (viii)  Matrix A has two rows and three columns

So, its order = number of rows x number of columns = 2-by-3.

Question:

Which of the following matrices are equal?

 A = $\left[ 3 \right]$ B =$~\left[ \begin{matrix} 3 & 5 \\ \end{matrix} \right]$ C = $\left[ 5-2 \right]$ D = $\left[ \begin{matrix} 5 & 3 \\ \end{matrix} \right]$ E = $~\left[ \begin{matrix} 4 & 0 \\ 6 & 2 \\ \end{matrix} \right]$ F = $\left[ \begin{matrix} 2 \\ 6 \\ \end{matrix} \right]$ G = $\left[ \begin{matrix} 3-1 \\ 3+3 \\ \end{matrix} \right]$ H = $\left[ \begin{matrix} 4 & 0 \\ 6 & 2 \\ \end{matrix} \right]$ I = $\left[ \begin{matrix} 3 & 3+2 \\ \end{matrix} \right]$ J = $\left[ \begin{matrix} 2+2 & 2-2 \\ 2+4 & 2+0 \\ \end{matrix} \right]$

Difficulty: Easy

Solution:

Solving C

C = $\left[ 5-2 \right]$

C = $\left[ 3 \right]$

Solving G

G = $\left[ \begin{matrix} 3-1 \\ 3+3 \\ \end{matrix} \right]$

G = $\left[ \begin{matrix} 2 \\ 6 \\ \end{matrix} \right]$

Solving I

I = $\left[ \begin{matrix} 3 & 3+2 \\ \end{matrix} \right]$

I = $\left[ \begin{matrix} 3 & 5 \\ \end{matrix} \right]$

Solving J

J = $\left[ \begin{matrix} 2+2 & 2-2 \\ 2+4 & 2+0 \\ \end{matrix} \right]$

J = $\left[ \begin{matrix} 4 & 0 \\ 6 & 2 \\ \end{matrix} \right]$

Now Matrices are said to be equal if

(i) They are of same order

(ii) Their corresponding values are equal

So, according to this definition

(a) Matrices A and C are equal, A = C.

(b) Matrices B and I are equal, B = I.

(c) Matrices E, H and J are equal, E = H = J.

(d) Matrices F and G are equal, F = G.

Question:

Find the values of a, b, c, and d which satisfy the matrix equation.

$\left[ \begin{matrix} a+c & a+2b \\ c-1 & 4d-6 \\ \end{matrix} \right] = \left[ \begin{matrix} 0 & -7 \\ 3 & 2d \\ \end{matrix} \right]$

Difficulty: Easy

Solution:

As, $\left[ \begin{matrix} a+c & a+2b \\ c-1 & 4d-6 \\ \end{matrix} \right]$ = $\left[ \begin{matrix} 0 & -7 \\ 3 & 2d \\ \end{matrix} \right]$

By comparing the corresponding elements, we get

$a + c = 0$

$a = -c$ ---------------(i)

$a + 2b = -7$

$2b = - (a+7)$ ---------------(ii)

$c - 1 = 3$

$c = 3 + 1$

$c = 4$ ---------------(iii)

By putting the value of “c” in equation (i), we will get

$a = -4$ ---------------(iv)

By putting the value of “a” in equation (ii), we will get

$2b = - (-4+7)$

$2b = - (3)$

$b = -{\large \frac{3}{2}}$

$b = - 1.5$ ---------------(v)

Similarly,

$4d - 6 = 2d$

$4d - 2d = 6$

$2d = 6$

$d =-{\large \frac{6}{2}}$

$d = 3$ ---------------(vi)

From equations (iii), (iv), (v) and (vi) we get

$a = - 4$, $b = - 1.5$, $c = 4$ and $d = 3$  