Construct a 4 × 3 matrix whose elements are given by a i j = i j .

Let A be the required matrix. It is a 4 × 3 matrix. So it has 4 rows and 3 columns. Given a i j = i j . So, a 11 = 1 1 = 1 , a 12 = 1 2 , a 13 = 1 3 a 21 = 2 1 = 2 , a 22 = 2 2 = 1 , a 23 = 2 3 , a 31 = 3 1 = 3 , a 32 = 3 2 , a 33 = 3 3 = 1 , a 41 = 4 1 = 4 , a 42 = 4 2 = 2 , a 43 = 4 3 In general, a matrix of order 4 × 3 is given by A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a 42 a 43 . So, the required matrix is A = 1 1 2 1 3 2 1 2 3 3 3 2 1 4 2 4 3 .

Construct a 4 × 3 matrix whose elements are given by a i j = i j .

Let A be the required matrix. It is a 4 × 3 matrix. So it has 4 rows and 3 columns. Given a i j = i j . So, a 11 = 1 1 = 1 , a 12 = 1 2 , a 13 = 1 3 a 21 = 2 1 = 2 , a 22 = 2 2 = 1 , a 23 = 2 3 , a 31 = 3 1 = 3 , a 32 = 3 2 , a 33 = 3 3 = 1 , a 41 = 4 1 = 4 , a 42 = 4 2 = 2 , a 43 = 4 3 In general, a matrix of order 4 × 3 is given by A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a 42 a 43 . So, the required matrix is A = 1 1 2 1 3 2 1 2 3 3 3 2 1 4 2 4 3 .

Construct a 2 × 2 matrix whose elements are a i j = i + 2 j 2 2 .

Let A be the required matrix. It is a 2 × 2 matrix. So, it has 2 rows and 2 columns. Given a i j = i + 2 j 2 2 So, a 11 = ( 1 + 2 × 1 ) 2 2 = 9 2 , a 12 = ( 1 + 2 × 2 ) 2 2 = 25 2 , a 21 = ( 2 + 2 × 1 ) 2 2 = 16 2 = 8 , a 22 = ( 2 + 2 × 2 ) 2 2 = 36 2 = 18 In general matrix of 2 × 2 order is given by A = a 11 a 12 a 21 a 22 . So, the required matrix is A = 9 2 25 2 8 18 .

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

Let A be the required matrix. It is a 2 × 3 matrix. So, it has 2 rows and 3 columns. Given a i j = i - 2 j 2 2 So, a 11 = ( 1 - 2 × 1 ) 2 2 = 1 2 a 12 = ( 1 - 2 × 2 ) 2 2 = 9 2 a 13 = ( 1 - 2 × 3 ) 2 2 = 25 2 a 21 = ( 2 - 2 × 1 ) 2 2 = 0 a 22 = 2 - 2 × 2 2 2 = 4 2 = 2 a 23 = 2 - 2 × 3 2 2 = 16 2 = 8 In general, a matrix of 2 × 3 order given by A = a 11 a 12 a 13 a 21 a 22 a 23 . So, the required matrix is A = 1 2 9 2 25 2 0 2 8 .

Construct a 3 × 4 matrix whose elements are given by a i j = 1 2 - 3 i + j .

Let A be the required matrix. It is a 3 × 4 matrix. So, it has 3 rows and 4 columns. Given a i j = 1 2 - 3 i + j . So, a 11 = 1 - 3 × 1 + 1 2 = - 2 2 = 2 2 = 1 a 12 = 1 - 3 × 1 + 2 2 = - 1 2 = 1 2 = 1 2 a 13 = = 1 - 3 × 1 + 3 2 = 0 2 = 0 a 14 = = 1 - 3 × 1 + 4 2 = 1 2 = 1 2 = 1 2 a 21 = 1 - 3 × 2 + 1 2 = - 5 2 = 5 2 a 22 = 1 - 3 × 2 + 2 2 = - 4 2 = 4 2 = 2 a 23 = 1 - 3 × 2 + 3 2 = - 3 2 = 3 2 a 24 = 1 - 3 × 2 + 4 2 = - 2 2 = 2 2 = 1 a 31 = 1 - 3 × 3 + 1 2 = - 8 2 = 8 2 = 4 a 32 = 1 - 3 × 3 + 2 2 = - 7 2 = 7 2 a 33 = 1 - 3 × 3 + 3 2 = - 6 2 = 6 2 = 3 a 34 = 1 - 3 × 3 + 4 2 = - 5 2 = 5 2 In general , a matrix of order 3 × 4 is given by A = a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 . So, the required matrix is A = 1 1 2 0 1 2 5 2 2 3 2 1 4 7 2 3 5 2 .

If A = 2 - 3 5 - 1 0 3 and B = 3 2 - 2 4 - 3 1 verify that A + B = B + A .

Given matrices A = 2 - 3 5 - 1 0 3 and B = 3 2 - 2 4 - 3 1 Clearly, A and B are of same order 2 × 3 . So, A + B and B + A are both defined. Now, A + B = 2 - 3 5 - 1 0 3 + 3 2 - 2 4 - 3 1 A + B = 2 + 3 - 3 + 2 5 - 2 - 1 + 4 0 - 3 3 + 1 = 5 - 1 3 3 - 3 4 ......(i) and B + A = 3 2 - 2 4 - 3 1 + 2 - 3 5 - 1 0 3 B + A = 3 + 2 2 - 3 - 2 + 5 4 - 1 - 3 + 0 1 + 3 = 5 - 1 3 3 - 3 4 .....(ii) From (i) and (ii), we get A + B = B + A Hence, proved that A + B = B + A

If A = 3 5 - 2 0 6 - 1 , B = - 1 - 3 4 2 - 2 3 and C = 0 2 3 - 4 1 6 , verify that A + B + C = A + B + C .

We are given matrices A = 3 5 - 2 0 6 - 1 , B = - 1 - 3 4 2 - 2 3 and C = 0 2 3 - 4 1 6 Clearly, each one of A , B and C is a matrix of order 3 × 2 . So, A + B + C and A + B + C are both defined. The addition all matrices is possible as order of all matrices are similar. Now, A + B + C = 3 5 - 2 0 6 - 1 + - 1 - 3 4 2 - 2 3 + 0 2 3 - 4 1 6 = 2 2 2 2 4 2 + 0 2 3 - 4 1 6 = 2 4 5 - 2 5 8 .....(i) and A + B + C = 3 5 - 2 0 6 - 1 + - 1 - 3 4 2 - 2 3 + 0 2 3 - 4 1 6 = 3 5 - 2 0 6 - 1 + - 1 - 1 7 - 2 - 1 7 = 2 4 5 - 2 5 8 ......(ii) From (i) and (ii), we get ( A + B ) + C = A + ( B + C ) Hence, proved that ( A + B ) + C = A + ( B + C ) .

If A = 3 1 2 1 2 - 3 and B = - 2 0 4 5 - 3 2 , find ( 2 A – B ) .

Given matrices are A = 3 1 2 1 2 - 3 and B = - 2 0 4 5 - 3 2 . We need to find 2 A - B . Now, 2 A = 2 3 1 2 1 2 - 3 = 2 × 3 2 × 1 2 × 2 2 × 1 2 × 2 2 × - 3 2 A = 6 2 4 2 4 - 6 We know that 2 A - B is the difference of two matrices 2 A and B . Clearly, both matrices of order 2 × 3 , subtraction of two matrices can be done. 2 A - B = 6 2 4 2 4 - 6 - - 2 0 4 5 - 3 2 2 A - B = 6 + 2 2 - 0 4 - 4 2 - 5 4 + 3 - 6 - 2 ∴ 2 A - B = 8 2 0 - 3 7 - 8 .

Let A = 2 4 3 2 , B = 1 3 - 2 5 and C = - 2 5 3 4 , f ind A + 2 B .

Given A = 2 4 3 2 , B = 1 3 - 2 5 To find A + 2 B . Now, 2 B = 2 1 3 - 2 5 = 2 × 1 2 × 3 2 × - 2 2 × 5 2 B = 2 6 - 4 10 We know that A + 2 B is the addition of two matrices A and 2 B . Clearly, both matrices of order 2 × 2 , addition of two matrices can be done. A + 2 B = 2 4 3 2 + 2 6 - 4 10 A + 2 B = 2 + 2 4 + 6 3 - 4 2 + 10 = 4 10 - 1 12 ∴ A + 2 B = 4 10 - 1 12 .

Let A = 2 4 3 2 , B = 1 3 - 2 5 and C = - 2 5 3 4 , f ind B – 4 C .

Given A = 2 4 3 2 , B = 1 3 - 2 5 and C = - 2 5 3 4 . We need to find B - 4 C . Now, 4 C = 4 - 2 5 3 4 = 4 × - 2 4 × 5 4 × 3 4 × 4 4 C = - 8 20 12 16 We know that B - 4 C is the difference of two matrices B and 4 C . Clearly, both matrices of order 2 × 2 , subtraction of two matrices can be done. B - 4 C = 1 3 - 2 5 - - 8 20 12 16 B - 4 C = 1 - ( - 8 ) 3 - 20 - 2 - 12 5 - 16 ∴ B - 4 C = 9 - 17 - 14 - 11 .

E M B I B E

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 5 - Matrices>EXERCISE>Q 6

EASY

12th CBSE

IMPORTANT

Earn 100

Construct a $4 \times 3$ matrix whose elements are given by $a_{i j} = \frac{i}{j}$ .

Important Questions on Matrices

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 5 - Matrices>EXERCISE>Q 7

Construct a $2 \times 2$ matrix whose elements are $a_{i j} = \frac{{(i + 2 j)}^{2}}{2}$ .

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 5 - Matrices>EXERCISE>Q 8

Construct a $(2 \times 3)$ matrix whose elements are $a_{i j} = \frac{{(i - 2 j)}^{2}}{2}$ .

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 5 - Matrices>EXERCISE>Q 9

Construct a $3 \times 4$ matrix whose elements are given by $a_{i j} = \frac{1}{2} |- 3 i + j|$ .

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 5 - Matrices>EXERCISE>Q 1

If $A = [\begin{matrix} 2 & - 3 & 5 \\ - 1 & 0 & 3 \end{matrix}]$ and $B = [\begin{matrix} 3 & 2 & - 2 \\ 4 & - 3 & 1 \end{matrix}]$ verify that $(A + B) = (B + A)$ .

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 5 - Matrices>EXERCISE>Q 2

If $A = [\begin{matrix} 3 & 5 \\ - 2 & 0 \\ 6 & - 1 \end{matrix}], B = [\begin{matrix} - 1 & - 3 \\ 4 & 2 \\ - 2 & 3 \end{matrix}]$ and $C = [\begin{matrix} 0 & 2 \\ 3 & - 4 \\ 1 & 6 \end{matrix}]$ , verify that $(A + B) + C = A + (B + C)$ .

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 5 - Matrices>EXERCISE>Q 3

If $A = [\begin{matrix} 3 & 1 & 2 \\ 1 & 2 & - 3 \end{matrix}]$ and $B = [\begin{array}{r} - 2 & 0 & 4 \\ 5 & - 3 & 2 \end{array}]$ , find $(2 A - B) .$

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 5 - Matrices>EXERCISE>Q 4

Let $A = [\begin{array}{l} 2 & 4 \\ 3 & 2 \end{array}], B = [\begin{matrix} 1 & 3 \\ - 2 & 5 \end{matrix}]$ and $C = [\begin{matrix} - 2 & 5 \\ 3 & 4 \end{matrix}]$ , find $A + 2 B$ .

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 5 - Matrices>EXERCISE>Q 4

Let $A = [\begin{array}{l} 2 & 4 \\ 3 & 2 \end{array}], B = [\begin{matrix} 1 & 3 \\ - 2 & 5 \end{matrix}]$ and $C = [\begin{matrix} - 2 & 5 \\ 3 & 4 \end{matrix}]$ , find $B - 4 C$ .