Construct a 3 × 3 matrix whose elements are a i j = i 2 j 2

The general 3 × 3 matrix is given by A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a i j = i 2 j 2 a 11 = 1 2 × 1 2 = 1 × 1 = 1 ; a 12 = 1 2 × 2 2 = 1 × 4 = 4 ; a 13 = 1 2 × 3 2 = 1 × 9 = 9 ; a 21 = 2 2 × 1 2 = 4 × 1 = 4 ; a 22 = 2 2 × 2 2 = 4 × 4 = 16 ; a 23 = 2 2 × 3 2 = 4 × 9 = 36 ; a 31 = 3 2 × 1 2 = 9 × 1 = 9 ; a 32 = 3 2 × 2 2 = 9 × 4 = 36 ; a 33 = 3 2 × 3 2 = 9 × 9 = 81 Hence, the required matrix is A = 1 4 9 4 16 36 9 36 81 .

Construct a 3 × 3 matrix whose elements are a i j = i 2 j 2

The general 3 × 3 matrix is given by A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a i j = i 2 j 2 a 11 = 1 2 × 1 2 = 1 × 1 = 1 ; a 12 = 1 2 × 2 2 = 1 × 4 = 4 ; a 13 = 1 2 × 3 2 = 1 × 9 = 9 ; a 21 = 2 2 × 1 2 = 4 × 1 = 4 ; a 22 = 2 2 × 2 2 = 4 × 4 = 16 ; a 23 = 2 2 × 3 2 = 4 × 9 = 36 ; a 31 = 3 2 × 1 2 = 9 × 1 = 9 ; a 32 = 3 2 × 2 2 = 9 × 4 = 36 ; a 33 = 3 2 × 3 2 = 9 × 9 = 81 Hence, the required matrix is A = 1 4 9 4 16 36 9 36 81 .

Find the value of a , b , c , d from the equation a - b 2 a + c 2 a - b 3 c + d = 1 5 0 2

The given matrices are equal. Thus, all the corresponding elements are equal. Therefore, a - b = 1 . . . 1 2 a + c = 5 . . . 2 2 a - b = 0 . . . 3 3 c + d = 2 . . . 4 3 gives 2 a - b = 0 2 a = b . . . 5 Put 2 a = b in equation 1 , a - 2 a = 1 gives a = - 1 Put a = - 1 in equation 5 , 2 - 1 = b gives b = - 2 Put a = - 1 in equation 2 , 2 - 1 + c = 5 gives c = 7 Put c = 7 in equation 4 , 3 7 + d = 2 gives d = - 19 Therefore, a = - 1 , b = - 2 , c = 7 , d = - 19 .

If A = 1 2 3 4 5 6 7 8 9 , B = 1 7 0 1 3 1 2 4 0 , find A + B .

Two matrices can be added or subtracted if they have the same order. To add or subtract two matrices, simply add or subtract the corresponding elements. Thus, we get, A + B = 1 2 3 4 5 6 7 8 9 + 1 7 0 1 3 1 2 4 0 = 1 + 1 2 + 7 3 + 0 4 + 1 5 + 3 6 + 1 7 + 2 8 + 4 9 + 0 = 2 9 3 5 8 7 9 12 9 . Therefore, the required sum is 2 9 3 5 8 7 9 12 9 .

If A = 1 3 - 2 5 - 4 6 - 3 2 9 , B = 1 8 3 4 9 6 , find A + B .

Two matrices can be added or subtracted if they have the same order. Here, we observe that the order of matrix A is 3 × 3 and that of matrix B is 3 × 2 . Hence, the order of matrix A and matrix B are not the same. Therefore, it is not possible to add A and B .

If A = 7 8 6 1 3 9 - 4 3 - 1 , B = 4 11 - 3 - 1 2 4 7 5 0 then find 2 A + B .

We can multiply the elements of the given matrix A by a non-zero number k to obtain a new matrix k A whose elements are multiplied by k . The matrix k A is called the scalar multiplication of A . Thus, if A = a i j m × n then, k A = k a i j m × n for all i = 1 , 2 , … , m and for all j = 1 , 2 , … , n . Two matrices can be added or subtracted if they have the same order. To add or subtract two matrices, simply add or subtract the corresponding elements. Since A and B have same order 3 × 3 , 2 A + B is defined. We have, 2 A + B = 2 7 8 6 1 3 9 - 4 3 - 1 + 4 11 - 3 - 1 2 4 7 5 0 = 14 16 12 2 6 18 - 8 6 - 2 + 4 11 - 3 - 1 2 4 7 5 0 = 18 27 9 1 8 22 - 1 11 - 2 Therefore, the required answer is 18 27 9 1 8 22 - 1 11 - 2 .

If A = 5 4 - 2 1 2 3 4 2 1 9 4 , B = - 7 4 - 3 1 4 7 2 3 5 - 6 9 , find 4 A - 3 B .

We can multiply the elements of the given matrix A by a non-zero number k to obtain a new matrix k A whose elements are multiplied by k . The matrix k A is called the scalar multiplication of A . Thus, if A = a i j m × n then, k A = k a i j m × n for all i = 1 , 2 , … , m and for all j = 1 , 2 , … , n . Two matrices can be added or subtracted if they have the same order. To add or subtract two matrices, simply add or subtract the corresponding elements. Since A , B are of the same order 3 × 3 , subtraction of 4 A and 3 B is defined. 4 A - 3 B = 4 5 4 - 2 1 2 3 4 2 1 9 4 - 3 - 7 4 - 3 1 4 7 2 3 5 - 6 9 = 20 16 - 8 2 3 4 2 4 36 16 + 21 - 12 9 - 3 4 - 21 2 - 9 - 15 18 - 27 = 41 4 1 5 4 - 15 2 4 2 - 9 - 11 54 - 11 Therefore, the required answer is 41 4 1 5 4 - 15 2 4 2 - 9 - 11 54 - 11 .

Find the value of a , b , c , d , x , y from the following matrix equation. d 8 3 b a + 3 a - 2 - 4 = 2 2 a b 4 c + 0 1 - 5 0

First, we add the two matrices on both left, right hand sides to get d + 3 8 + a 3 b - 2 a - 4 = 2 2 a + 1 b - 5 4 c . [Two matrices can be added or subtracted if they have the same order. To add or subtract two matrices, simply add or subtract the corresponding elements.] Equating the corresponding elements of the two matrices, we have d + 3 = 2 gives d = - 1 8 + a = 2 a + 1 gives a = 7 3 b - 2 = b - 5 gives b = - 3 2 Substituting a = 7 in a - 4 = 4 c gives c = 3 4 Therefore, a = 7 , b = - 3 2 , c = 3 4 , d = - 1

If A = 1 8 3 3 5 0 8 7 6 , B = 8 - 6 - 4 2 11 - 3 0 1 5 , C = 5 3 0 - 1 - 7 2 1 4 3 compute the following 3 A + 2 B - C .

We can multiply the elements of the given matrix A by a non-zero number k to obtain a new matrix k A whose elements are multiplied by k . The matrix k A is called the scalar multiplication of A . Thus, if A = a i j m × n then, k A = k a i j m × n for all i = 1 , 2 , … , m and for all j = 1 , 2 , … , n . Two matrices can be added or subtracted if they have the same order. To add or subtract two matrices, simply add or subtract the corresponding elements. 3 A + 2 B - C = 3 1 8 3 3 5 0 8 7 6 + 2 8 - 6 - 4 2 11 - 3 0 1 5 - 5 3 0 - 1 - 7 2 1 4 3 = 3 24 9 9 15 0 24 21 18 + 16 - 12 - 8 4 22 - 6 0 2 10 + - 5 - 3 0 1 7 - 2 - 1 - 4 - 3 = 14 9 1 14 44 - 8 23 19 25 Therefore, the required answer is 14 9 1 14 44 - 8 23 19 25 .

E M B I B E

Mathematics Standard 10>Chapter 3 - Algebra>Solved Example>Q 56

MEDIUM

10th Tamil Nadu Board

IMPORTANT

Earn 100

Construct a $3 \times 3$ matrix whose elements are $a_{i j} = i^{2} j^{2}$

Important Questions on Algebra

MEDIUM

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 3 - Algebra>Solved Example>Q 57

Find the value of $a, b, c, d$ from the equation $(\begin{matrix} a - b & 2 a + c \\ 2 a - b & 3 c + d \end{matrix}) = (\begin{matrix} 1 & 5 \\ 0 & 2 \end{matrix})$

MEDIUM

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 3 - Algebra>Solved Example>Q 58

If $A = (\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}), B = (\begin{matrix} 1 & 7 & 0 \\ 1 & 3 & 1 \\ 2 & 4 & 0 \end{matrix}),$ find $A + B$ .

MEDIUM

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 3 - Algebra>Solved Example>Q 59

Two examinations were conducted for three groups of students namely group $1$ , group $2$ , group $3$ and their data on average of marks for the subjects Tamil, English, Science and Mathematics are given below in the form of matrices $A$ and $B$ . Find the total marks of both the examinations for all three groups.

$\begin{matrix} Tamil & English & Science & Mathematics \end{matrix} A = \begin{matrix} Group 1 \\ Group 2 \\ Group 3 \end{matrix} (\begin{matrix} 22 & 15 & 14 & 23 \\ 50 & 62 & 21 & 30 \\ 53 & 80 & 32 & 40 \end{matrix})$

$\begin{matrix} Tamil & English & Science & Mathematics \end{matrix} B = \begin{matrix} Group 1 \\ Group 2 \\ Group 3 \end{matrix} (\begin{matrix} 20 & 38 & 15 & 40 \\ 18 & 12 & 17 & 80 \\ 81 & 47 & 52 & 18 \end{matrix})$

EASY

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 3 - Algebra>Solved Example>Q 60

If $A = (\begin{matrix} 1 & 3 & - 2 \\ 5 & - 4 & 6 \\ - 3 & 2 & 9 \end{matrix}), B = (\begin{matrix} 1 & 8 \\ 3 & 4 \\ 9 & 6 \end{matrix}),$ find $A + B$ .

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 3 - Algebra>Solved Example>Q 61

If $A = (\begin{matrix} 7 & 8 & 6 \\ 1 & 3 & 9 \\ - 4 & 3 & - 1 \end{matrix}), B = (\begin{matrix} 4 & 11 & - 3 \\ - 1 & 2 & 4 \\ 7 & 5 & 0 \end{matrix})$ then find $2 A + B$ .

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 3 - Algebra>Solved Example>Q 62

If $A = (\begin{matrix} 5 & 4 & - 2 \\ \frac{1}{2} & \frac{3}{4} & \sqrt{2} \\ 1 & 9 & 4 \end{matrix}), B = (\begin{matrix} - 7 & 4 & - 3 \\ \frac{1}{4} & \frac{7}{2} & 3 \\ 5 & - 6 & 9 \end{matrix})$ , find $4 A - 3 B$ .

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 3 - Algebra>Solved Example>Q 63

Find the value of $a, b, c, d, x, y$ from the following matrix equation.

$(\begin{matrix} d & 8 \\ 3 b & a \end{matrix}) + (\begin{matrix} 3 & a \\ - 2 & - 4 \end{matrix}) = (\begin{matrix} 2 & 2 a \\ b & 4 c \end{matrix}) + (\begin{matrix} 0 & 1 \\ - 5 & 0 \end{matrix})$

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 3 - Algebra>Solved Example>Q 64

If $A = (\begin{matrix} 1 & 8 & 3 \\ 3 & 5 & 0 \\ 8 & 7 & 6 \end{matrix}), B = (\begin{matrix} 8 & - 6 & - 4 \\ 2 & 11 & - 3 \\ 0 & 1 & 5 \end{matrix}), C = (\begin{matrix} 5 & 3 & 0 \\ - 1 & - 7 & 2 \\ 1 & 4 & 3 \end{matrix})$ compute the following $3 A + 2 B - C$ .