Please Wait... While Loading Full Video#### Class 12 Chapter 3 - Matrices

### MATRICES FOR CBSE - NCERT

`star` Matrix

`star` Types of matrix

`star` Types of matrix

`\color{green} ✍️` A matrix is an ordered rectangular array of numbers or functions.

`\color{green} ✍️` The numbers or functions are called the elements or the entries of the matrix

`\color{green} ✍️` We denote matrices by capital letters. The following are some examples of matrices:

● In the examples, the horizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix.

●Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2 rows and 3 columns.

`\color{green} ✍️` The numbers or functions are called the elements or the entries of the matrix

`\color{green} ✍️` We denote matrices by capital letters. The following are some examples of matrices:

● In the examples, the horizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix.

●Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2 rows and 3 columns.

`color{green} ✍️` A matrix having `m` rows and `n` columns is called a matrix of order `color{red}{m × n}` or simply `m × n` matrix (read as an m by n matrix).

`color{green} ✍️` In Fig the examples of matrices is shown

`color{green} ✍️`, we have `A = [(-2,5,1),(1,2,9)]` as `color{green}{2× 3}`

Matrix `B = [(1,2,3),(4,5,6),(7,8,9)]` as `3 × 3` matrix.

`color{green} ✍️` Thus the `i^(th)` row consist of the elements `a_(i1),a_(i2),a_(i3),..........,a_(in),` while the `j^(th)` column consists of the elements `a_(1j),a_(2j),a_(3j),..........,a_(mj),`

`color{blue}{Note :}` 1. We shall follow the notation, namely `A = [a_ij]_(m × n)` to indicate that `A` is a matrix of order `m × n.`

2. We shall consider only those matrices whose elements are real numbers or functions taking real values.

`color{green} ✍️` In Fig the examples of matrices is shown

`color{green} ✍️`, we have `A = [(-2,5,1),(1,2,9)]` as `color{green}{2× 3}`

Matrix `B = [(1,2,3),(4,5,6),(7,8,9)]` as `3 × 3` matrix.

`color{green} ✍️` Thus the `i^(th)` row consist of the elements `a_(i1),a_(i2),a_(i3),..........,a_(in),` while the `j^(th)` column consists of the elements `a_(1j),a_(2j),a_(3j),..........,a_(mj),`

`color{blue}{Note :}` 1. We shall follow the notation, namely `A = [a_ij]_(m × n)` to indicate that `A` is a matrix of order `m × n.`

2. We shall consider only those matrices whose elements are real numbers or functions taking real values.

Q 3174856756

Consider the following information regarding the number of men and women workers in three factories I, II and III

Represent the above information in the form of a `3 × 2` matrix. What does the entry in the third row and second column represent?

Represent the above information in the form of a `3 × 2` matrix. What does the entry in the third row and second column represent?

The information is represented in the form of a `3 × 2` matrix as follows :

`A = [(30,25),(25,31),(27,26)]`

Q 3174056856

If a matrix has 8 elements, what are the possible orders it can have?

We know that if a matrix is of order `m × n`, it has `mn` elements. Thus, to find all possible orders of a matrix with 8 elements, we will find all ordered pairs of natural numbers, whose product is `8.`

Thus, all possible ordered pairs are `(1, 8), (8, 1), (4, 2), (2, 4)`

Hence, possible orders are `1 × 8, 8 ×1, 4 × 2, 2 × 4`

Q 3174156956

Construct a `3 × 2` matrix whose elements are given by `a_(ij) = 1/2 |i-3j|`

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 |i - 3j|, i=1,2,3 ` and `j=1,2`

Therefore `a_11 = 1/2 |1-3 xx 1| =1 \ a_12 =1/2 |1-3xx2| = 5/2`

`a_21 = 1/2 | 2- 3xx 1|=1/2 \ a_22 = 1/2 | 2-3xx2|=2`

`a_31 = 1/2 |3 -3 xx 1|=0 \ a_32 = 1/2 |3 -3 xx 2|=3/2`

Hence the required matrix is given by `A= [(1, 5/2) , (1/2 , 2), (0,3/2)]`

`\color{green} ✍️` A matrix is said to be a column matrix if it has only one column.

`\color{green} ✍️` For example, `A = [(0,,),(3,,),(-1,,),(4,,)]` is a column matrix of order 4 × 1.

`color{green}{"In general, If n= 1 for column matrix or " A = [a_(ij)]_(m xx 1) "is a column matrix of order m × 1"}`

`\color{green} ✍️` For example, `A = [(0,,),(3,,),(-1,,),(4,,)]` is a column matrix of order 4 × 1.

`color{green}{"In general, If n= 1 for column matrix or " A = [a_(ij)]_(m xx 1) "is a column matrix of order m × 1"}`

`\color{green} ✍️` A matrix is said to be a row matrix if it has only one row.

Example : `[ 0 \ \ -4 \ \ 3 \ \ 1]`

`color{green}{"In general, "B = [b_(ij)]_(1 xxn) "is a row matrix of order" 1 × n.}`

Example : `[ 0 \ \ -4 \ \ 3 \ \ 1]`

`color{green}{"In general, "B = [b_(ij)]_(1 xxn) "is a row matrix of order" 1 × n.}`

`\color{green} ✍️` A matrix in which the number of `"rows"` are equal to the number of `"columns"`, is said to be a square matrix.

Thus an `m × n` matrix is said to be a square matrix if `m = n` and is known as a square matrix of order ‘m’.

examples : `[(1, -1, 0),(8,2,3),(-1,-2,6)]`

`color{green}{"In general, "A = [a_(ij)]_(mxx m)" is a square matrix of order m."}`

Thus an `m × n` matrix is said to be a square matrix if `m = n` and is known as a square matrix of order ‘m’.

examples : `[(1, -1, 0),(8,2,3),(-1,-2,6)]`

`color{green}{"In general, "A = [a_(ij)]_(mxx m)" is a square matrix of order m."}`

`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)]`

`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)]`

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

`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)]`

`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)]`

`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)]`

`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)]`

`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)]`

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

`\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`

(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

Find the values of a, b, c, x, y and z.

Class 12 Chapter 3 Example 4

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

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

Class 12 Chapter 3 Example 5

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