Important Questions for Class 12 Chapter 3 - Matrices

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Important questions based on NCERT syllabus for Class 12 Chapter 3 - Matrices:

Question 1: Construct a 3 × 2 matrix whose elements are given by a_{ij} = 1/2 |2i - j|

Solution: In general a 3 × 2 matrix is given by A = a_{11} a_{12}
a_{21} a_{22}
a_{31} a_{32}

Now, a_{ij} = 1/2 |2i - j|, i = 1,2,3 and j = 1, 2

Therefore, a_{11} = 1/2 |2(1) - 1| = 1/2|1| = 1/2
a_{12} = 1/2 |2(1) - 2| = 1/2|0| = 0
a_{21} = 1/2 |2(2) - 1| = 1/2|3| = 3/2
a_{22} = 1/2 |2(2) - 2| = 1/2|2| = 1
a_{31} = 1/2 |2(3) - 1| = 1/2|5| = 5/2
a_{32} = 1/2 |2(3) - 2| = 1/2 | 6 - 2| = 1/2|4| = 1/2

Hence the required matrix is given by
A = 1/2 0
3/2 1
5/2 1/2

Question 2: Write the value of x + 2y - z from the following equation

x + y - z 7
x - z 8
y + z 10

Solution: By definition of equality of matrices i.e., if two matrices are equal then their corresponding elements are equal.

Given
x + y - z 7
x - z 8
y - z 10

on equating the corresponding elements we get,

x + y - z = 7 ..........(1)
x - z = 8........(2)
y - z = 10 ......(3)

Consider equation 1 we get

x + y- z = 7
we can group it as x + (y - z) = 7.....(4)

we know that y - z = 10

Substitute the equation 3 in equation 4 we get

x + (10) = 7 => x = -3

Substitute x value in equation 2 to get value of z

x - z = 8 => -3 - z = 8 => z = -11

Substitute z value in equation 3 to get y value

y - z = 10 => y -(-11) = 10 => y = 10 - 11=> y = -1

Now we have to find the value of x + 2y - z
we know x = -3, y = -1 and z = -11

x + 2y - z = -3 + 2(-1) - 11 = -3 - 2 - 11 = -16

Question 3: If A = -2 4 8 B = 5 6
6 1 0 8 0
1 3
Find AB and BA And prove that $AB \neq BA$

Solution: A = -2 4 8 B = 5 6
6 1 0 8 0
1 3
AB = -10+32+8 -12+0+24
30+8+0 36+0+0

AB = 30 12
38 36

BA = 5 6 -2 4 8
8 0 6 1 0
1 3

= -10+36 20+6 40+0
-16+0 32+0 64+0
-2+18 4+3 8+0

= 26 26 40
-16 32 64
16 7 8

Hence $AB \neq BA$ = 30 12 \neq 26 26 40
38 36 -16 32 64
16 7 8