Mathematics 9-Ch1 Matrices

Table of Contents

• Order of the following matrices
• Which of the following matrices are equal
• Find the values of a, b, c, and d which satisfy the matrix equation

Question:

 

Find the order of the following matrices.

 

A = $\left[\begin{array}{cc}2& 3\\ -5& 6\end{array}\right]$	B = $\left[\begin{array}{cc}2& 0\\ 3& 5\end{array}\right]$
C = $\left[\begin{array}{cc}2& 4\end{array}\right]$	D = $\left[\begin{array}{c}4\\ 0\\ 6\end{array}\right]$
E = $\left[\begin{array}{cc}�& �\\ �& �\\ �& �\end{array}\right]$	F = $\left[2\right]$
G=$\left[\begin{array}{ccc}2& 3& 0\\ 1& 2& 3\\ 2& 4& 5\end{array}\right]$	H = $\left[\begin{array}{ccc}2& 3& 4\\ 1& 0& 6\end{array}\right]$

 

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

 

Solution:

Solving C

C = $[5-2]$

C = $\left[3\right]$

 

Solving G

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

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

 

Solving I

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

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

 

Solving J

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

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

 

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.

 

$\left[\begin{array}{cc}�+�& �+2�\\ �-1& 4�-6\end{array}\right]=\left[\begin{array}{cc}0& -7\\ 3& 2�\end{array}\right]$

 

 

Solution:

As, $\left[\begin{array}{cc}�+�& �+2�\\ �-1& 4�-6\end{array}\right]$ = $\left[\begin{array}{cc}0& -7\\ 3& 2�\end{array}\right]$

 

By comparing the corresponding elements, we get

$�+�=0$

$�=-�$ ---------------(i)

 

$�+2�=-7$

$2�=-(�+7)$ ---------------(ii)

 

$�-1=3$

$�=3+1$

$�=4$ ---------------(iii)

 

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

$�=-4$ ---------------(iv)

 

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

$2�=-(-4+7)$

$2�=-(3)$

$�=-\frac{3}{2}$

$�=-1.5$ ---------------(v)

 

Similarly,

$4�-6=2�$

$4�-2�=6$

$2�=6$

$�=-\frac{6}{2}$

$�=3$ ---------------(vi)

 

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

$�=-4$, $�=-1.5$, $�=4$ and $�=3$