Multiplication MCQ Quiz in मल्याळम - Objective Question with Answer for Multiplication - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Explanation:

Given:

, and 

While

Thus AB = 0 and BA ≠ 0.

Additional Information

∵ AB ≠ BA

⇒ Matrix A and B do not commute. 

Concept:

Identity matrix:

An identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros.

The effect of multiplying a given matrix by an identity matrix is to leave the given matrix unchanged.

Calculation:

Given:

A = 

Therefore, A = 3I

As we know, I3 = I, I2 = I               (∵ Identity matrix is a unit matrix)

Now,

A3 = (3I)3 = 27I3 = 27I 

⇒ A³ = 9 × 3I

∴ A³ = 9A                         (∵ A = 3I)

Concept:

Multiplication of matrices:

• The number of columns of the 1st matrix must equal the number of rows of the 2nd matrix.
• And the result will have the same number of rows as the 1st matrix, and the same number of columns as the 2nd matrix.
• To multiply an m × n matrix by an n × p matrix, the n must be the same, and the result is an m × p matrix.

Assume C is the product of two matrices A and B,

⇒ C = AB

Let A = , B =  and C =

⇒

Here c_ik = a_ij b_jk

Calculation:

Given:

Let A =

⇒ Aⁿ = I

Now, A² = AA =

⇒ A² =

⇒ A² = I

Now,

If n/2 is multiple of 2 than I^n/2 = I

∴ n is even number.

Concept:

The transpose of a matrix is simply a flipped version of the original matrix. We can get transpose by switching its rows with its columns. it is denoted by AT.

Calculation:​

Given:   and  

AB = 

⇒ 

⇒ 

Now, we can get transpose of AB by switching its rows with its columns.

∴ (AB)^T = 

Hence, option (2) is correct. 

Concept:

Properties of matrix multiplication:

The commutative property of multiplication DOES NOT NECESSARILY HOLD!

AB ≠ BA

If AB = BA, then we say that A and B commute.

Associative property: (AB)C = A(BC)

Calculation:

We have, (A+B)(A−B)

= A(A−B) + B(A−B)  [∵ Matrix product is distributive.]

= A² − AB + BA −B²

Since the matrix product is not commutative, so AB ≠ BA

∴ (A+B)(A−B) = A2 – B2 + BA – AB

Given :

Concept :

Multiplication of matrix:

Calculations :

We have 

Then by using the above concept 

⇒ 

⇒ 

∴ Comparing both matrices, the value of λ is -7.

Given:

 and 

Concept:

Matrix Multiplication:

Multiplication is only possible when the number of columns of the first matrix is equal to the number of rows of the second matrix.

A m × n matrix multiplied by a n × p matrix results in a m × p matrix.

Calculation:

⇒     ----(1)

Similarly, 

    ----(2)

If AB = BA

⇒ 

This will be possible if

2b = 2q or 3a = 3b

⇒ a = b

∴ There exist infinitely many B's such that AB = BA

Concept:

Matrix Multiplication:

Multiplication is only possible when the number of columns of the first matrix is equal to the number of rows of the second matrix.

A m × n matrix multiplied by a n × p matrix results in a m × p matrix.

Matrices are multiplied by multiplying each element of a row of the first m×n matrix with the corresponding elements of all the columns of the second n×p matrix to obtain the first row of the product matrix with p columns, and so on for all the m rows of the first matrix.

Calculation:

We have,

⇒ 

⇒ 

⇒ 1 + 2x + 2 + x + 3 = 0

⇒ 3x + 6 = 0

⇒ x = -2

∴ Required value of x is -2.

Concept:

Multiplication of 2 x 2 matrices:

AB =  = 

Calculation:

Given

A = 

A² =  = 

A2 = 

Concept:

Matrix Multiplication:

• Two matrices can only be multiplied when the column number of the first matrix is equal to the row number of the second matrix
• If A is not an invertible matrix then |A| = 0
• If A is a singular matrix then |A| = 0

Calculation:

Given that,

⇒ |A| = 1(1 - 1) - 1(1 - 1) + 1 (1 - 1) = 0

Hence, Inverse of A does not exist.

⇒ A² = 

⇒ A² = 

⇒ A³ = A² . A

⇒ A³ = 

⇒ A³ = 

⇒ A³ = 9 ≠ A

Hence, statement (2) is incorrect.

Now, the value of 3A is

= 3 

⇒ 3A = 

∴  3A = A²

Hence, statement (3) is correct.