Find a matrix A such that 2 A - 3 B + 5 C = 0 , where B = - 2 2 0 3 1 4 and C = 2 0 - 2 7 1 6

We have, 2 A - 3 B + 5 C = O ⇒ 2 A = 3 B - 5 C ⇒ 2 A = 3 - 2 2 0 3 1 4 - 5 2 0 - 2 7 1 6 2 A = - 6 6 0 9 3 12 - 10 0 - 10 35 5 30 2 A = - 16 6 10 - 26 - 2 - 18 ⇒ A = - 8 3 5 - 13 - 1 - 9

If F x = cos x - sin x 0 sin x cos x 0 0 0 1 , then show that F x F y = F x + y

Given that, F x = cos x - sin x 0 sin x cos x 0 0 0 1 To proof : F x F y = F x + y Now, F x F y = cos x - sin x 0 sin x cos x 0 0 0 1 cos y - sin y 0 sin y cos y 0 0 0 1 ⇒ F x F y = cos x cos y - sin x sin y - cos x sin y + sin x cos y 0 sin x cos y + cos x sin y cos x cos y - sin x sin y 0 0 0 1 ⇒ F x F y = cos x + y - sin x + y 0 sin x + y cos x + y 0 0 0 1 . . . . . . . . . . . . . . . 1 ⇒ F x + y = cos x + y - sin x + y 0 sin x + y cos x + y 0 0 0 1 . . . . . . . . . . . . 2 From equation 1 and 2 , F x F y = F x + y Hence, proved.

Find X and Y , if X + Y = 7 0 2 5 and X - Y = 3 0 0 3

Given: X + Y = 7 0 2 5 ⋯ ⋯ ( 1 ) and X - Y = 3 0 0 3 ⋯ ⋯ ( 2 ) Adding equation ( 1 ) and ( 2 ) , we get X + Y + X - Y = 7 0 2 5 + 3 0 0 3 ⇒ 2 X = 7 + 3 0 + 0 2 + 0 5 + 3 2 X = 10 0 2 8 ⇒ X = 1 2 10 0 2 8 ⇒ X = 10 2 0 2 2 2 8 2 ⇒ X = 5 0 1 4 Putting the value of X in equation ( 1 ) , we get 5 0 1 4 + Y = 7 0 2 5 Y = 7 0 2 5 - 5 0 1 4 ⇒ Y = 7 - 5 0 - 0 2 - 1 5 - 4 ⇒ Y = 2 0 1 1 Therefore, X = 5 0 1 4 and Y = 2 0 1 1 .

Compute the indicated products. 1 2 3 2 3 4

To find: 1 2 3 2 3 4 Here order of the first matrix is 3 × 1 and the second matrix is 1 × 3 . As here the number of columns in the first matrix is same as number of rows in the second matrix, by using the multiplication principle of matrices we get, 1 2 3 2 3 4 = 1 × 2 1 × 3 1 × 4 2 × 2 2 × 3 2 × 4 3 × 2 3 × 3 3 × 4 = 2 3 4 4 6 8 6 9 12 Hence, 1 2 3 2 3 4 = 2 3 4 4 6 8 6 9 12

If A = cos α sin α - sin α cos α , then verify that A ' A = I

Given: A = cos α sin α - sin α cos α To get transpose interchange rows and columns. ⇒ A ' = cos α - sin α sin α cos α Therefore A ' A = cos α - sin α sin α cos α cos α sin α - sin α cos α = cos 2 α + sin 2 α cos α sin α - cos α sin α cos α sin α - cos α sin α cos 2 α + sin 2 α = 1 0 0 1 = I ∴ A ' A = I Hence, proved

MEDIUM

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Important Questions on Matrices

EASY

Mathematics>Algebra>Matrices>Basics of Matrices

When the same monochromatic ray of light travels through glass slab and through water, the number of waves in glass slab of thickness

6 cm

is same as in water column of height

7 cm

. If refractive index of glass is

1.5

then refractive index of water is

EASY

Mathematics>Algebra>Matrices>Basics of Matrices

Find a matrix

A

such that

2 A - 3 B + 5 C = 0

, where

B = [\begin{array}{r} - 2 & 2 & 0 \\ 3 & 1 & 4 \end{array}]

and

C = [\begin{array}{r} 2 & 0 & - 2 \\ 7 & 1 & 6 \end{array}]

MEDIUM

Mathematics>Algebra>Matrices>Basics of Matrices

F (x) = [\begin{matrix} \cos x & - \sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{matrix}]

, then show that

F (x) F (y) = F (x + y)

EASY

Mathematics>Algebra>Matrices>Basics of Matrices

Find $X$ and $Y$ , if $X + Y = [\begin{array}{l} 7 & 0 \\ 2 & 5 \end{array}]$ and $X - Y = [\begin{array}{l} 3 & 0 \\ 0 & 3 \end{array}]$

MEDIUM

Mathematics>Algebra>Matrices>Basics of Matrices

Let

A = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}], (a \in R)

such that

A^{32} = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] .

Then, a value of

α

is:

EASY

Mathematics>Algebra>Matrices>Basics of Matrices

Compute the indicated products.

$[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}] [\begin{array}{l} 2 & 3 & 4 \end{array}]$

MEDIUM

Mathematics>Algebra>Matrices>Basics of Matrices

A

is a square matrix such that

A^{2} = A

, then

(I - A)^{3} + A

is equal to

EASY

Mathematics>Algebra>Matrices>Basics of Matrices

FInd the number of all possible matrices of order

$3 \times 3$ with each entry $0$ or $1$ .

EASY

Mathematics>Algebra>Matrices>Basics of Matrices

A

is a symmetric matrix and

B

is skew- symmetric matrix such that

A + B = [\begin{matrix} 2 & 3 \\ 5 & - 1 \end{matrix}]

, then

A B

is equal to:

MEDIUM

Mathematics>Algebra>Matrices>Basics of Matrices

A

is a square matrix such that

A^{2} = I

, then

(A - I)^{3} + (A + I)^{3} - 7 A

is equal to

EASY

Mathematics>Algebra>Matrices>Basics of Matrices

A = [\begin{array}{l} \cos α & \sin α \\ - \sin α & \cos α \end{array}]

, then verify that

A^{'} A = I

MEDIUM

Mathematics>Algebra>Matrices>Basics of Matrices

A = [\begin{matrix} 0 \\ 1 \end{matrix} \begin{matrix} - 1 \\ 0 \end{matrix}]

, then which one of the following statements is not correct?

MEDIUM

Mathematics>Algebra>Matrices>Basics of Matrices

The number of possible matrices of order

3 \times 3

with each entry

0

1

MEDIUM

Mathematics>Algebra>Matrices>Basics of Matrices

A = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}]

, and

A + A^{'} = I

, then the value of

α

EASY

Mathematics>Algebra>Matrices>Basics of Matrices

A

is a matrix of order

3 \times 2

, and order of the matrix

A^{'}

m \times n

, then

m + n =

_____

EASY

Mathematics>Algebra>Matrices>Basics of Matrices

A

and

B

are two matrices such that

A B = B, B A = A

then

A^{2} + B^{2} =

MEDIUM

Mathematics>Algebra>Matrices>Basics of Matrices

If $B$ is a $3 \times 3$ matrix such that $B^{2} = 0$ , then $d e t . [{(I + B)}^{50} - 50 B]$ is equal to :

MEDIUM

Mathematics>Algebra>Matrices>Basics of Matrices

If $A = [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{matrix}]$ is a matrix satisfying the equation $A A^{T} = 9 I$ , where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to