Mathematics Matrix For CBSE-NCERT 2

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Class 12 Chapter 3 - Matrices

Matrix For CBSE-NCERT 2

Topic covered

`star` Types Of Matrix
`star` Equality Of Matrix

Types of matrix

● Diagonal matrix
● Scalar matrix
● Identity matrix
● Zero matrix

Diagonal matrix

`color{green} ✍️` A square matrix `B = [b_(ij)]_(m xx m)` is said to be a diagonal matrix if `color{green}{"all its non diagonal elements are zero"`,

`color{green} ✍️` A matrix `B = [b_(ij)] m xx m` is said to be a diagonal matrix `color{red } {if b_(ij) = 0, "when" \ \ i ≠ j`

E.g. `A = [4],` `B = [(-1,0),(0,-1)],` `c = [(-1,0,0),(0,1,0),(0,0,1)]`

Scalar matrix

`color{green} ✍️` A diagonal matrix `B = [b_(ij)]_(m xx m)` is said to be a scalar matrix if all its `color{green}{"non diagonal elements are zero,"}`

`color{green} ✍️` A matrix `B = [b_(ij)] m xx m` is said to be a scalar matrix if

`color{orange}{
b_(ij) = 0,\ \ "when" \ \ i ≠ j}`

`color{orange}{
b_(ij) = k "when" \ \ i=j}`

E.g. `A = [4],` `B = [(-1,0),(0,-1)],` `c = [(5,0,0),(0,5,0),(0,0,5)]`

Identity matrix

`color{green} ✍️` A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. In other words, the square matrix

`color{orange}{b_(ij) = 0, \ \ "when" \ \ i ≠ j}`
`color{orange}{b_(ij) = 1, \ \ "when" \ \ i=j}`

E.g. `A = [1],` `B = [(1,0),(0,1)],` `c = [(1,0,0),(0,1,0),(0,0,11)]`

Zero matrix

`color{green} ✍️` A matrix is said to be zero matrix or null matrix if all its elements are zero.

E.g. `A = [0],` `B = [(0,0),(0,0)],` `c = [(0,0,0),(0,0,0),(0,0,0)]`

Summery

Equality of matrices

`\color{green} ✍️` Two matrices `A = [a_(ij)]` and `B = [b_(ij)]` are said to be equal if

(i) they are of the same order
(ii) each element of A is equal to the corresponding element of B, that is `a_(ij) = b_(ij)` for all `i` and `j.`

`color{blue}{A = B =>[a_(ij)] = [b_(ij)]}`

if `[(x,y),(z,a),(b,c)] = [(-1,0),(2,sqrt6),(3,2)] ` then `x = -, y =0, z=2 ,a = sqrt6, b=3,c=2`

Q 3124578451

If ` [ (x+3, z+4, 2y-7 ), (-6,a-1,0 ), ( b-3, -21 ,0 ) ] = [ (0, 6, 3y-2 ), ( -6, -3, 2c+2), ( 2b+4, -21 , 0 ) ]`

Find the values of a, b, c, x, y and z.
Class 12 Chapter 3 Example 4

Solution:

As the given matrices are equal, therefore, their corresponding elements
must be equal. Comparing the corresponding elements, we get

x + 3 = 0, z + 4 = 6, 2y – 7 = 3y – 2
a – 1 = – 3, 0 = 2c + 2, b – 3 = 2b + 4,

Simplifying, we get

a = – 2, b = – 7, c = – 1, x = – 3, y = –5, z = 2

Q 3144578453

Find the values of a, b, c, and d from the following equation:

`[ (2a+b , a-2b ), ( 5c-d, 4c +3d) ] = [ (4,-3), ( 11,24 ) ]`
Class 12 Chapter 3 Example 5

Solution:

By equality of two matrices, equating the corresponding elements, we get
2a + b = 4, 5c – d = 11
a – 2b = – 3 , 4c + 3d = 24

Solving these equations, we get
a = 1, b = 2, c = 3 and d = 4