Matrix class 12th most important objective question for cbse and bseb board exam 2021

  

#Link of objective question is below the notes

Class 12 Maths Notes Chapter 3 Matrices

Matrix: A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

Order of a Matrix: If a matrix has m rows and n columns, then its order is written as m × n. If a matrix has order m × n, then it has mn elements.

In general, am×n matrix has the following rectangular array:

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

Types of Matrices
Column Matrix: A matrix which has only one column, is called a column matrix.
e.g. ⎡⎣⎢10−5⎤⎦⎥
In general, A = [aij]m×1 is a column matrix of order m × 1.

Row Matrix: A matrix which has only one row, is called a row matrix,
e.g. [159]
In general, A = [aij]1×n is a row matrix of order 1 x n. 

Square Matrix: A matrix which has equal number of rows and columns, is called a square matrix
e.g. [35−12]
In general, A = [aij]m x m is a square matrix of order m.
Note: If A = [aij] is a square matrix of order n, then elements a11, a22, a33,…, ann is said to constitute the diagonal of the matrix A.

Diagonal Matrix: A square matrix whose all the elements except the diagonal elements are zeroes, is called a diagonal matrix,
e.g. ⎡⎣⎢3000−3000−8⎤⎦⎥
In.general, A = [aij]m×m is a diagonal matrix, if aij = 0, when i ≠ j

Scalar Matrix: A diagonal matrix whose all diagonal elements are same (non-zero), is called a scalar matrix,
e.g. ⎡⎣⎢200020002⎤⎦⎥
In general, A = [aij]n×n is a scalar matrix, if aij = 0, when i ≠ j, aij = k (constant), when i = j.
Note: A scalar matrix is a diagonal matrix but a diagonal matrix may or may not be a scalar matrix.

Unit or Identity Matrix: A diagonal matrix in which all diagonal elements are ‘1’ and all non-diagonal elements are zero, is called an identity matrix. It is denoted by I.
e.g. ⎡⎣⎢100010001⎤⎦⎥
In general, A = [aij]n×n is an identity matrix, if aij = 1, when i = j and aij = 0, when i ≠ j.

Zero or Null Matrix: A matrix is said to be a zero or null matrix, if its all elements are zer0
e.g. [0000]

Equality of Matrices: Two matrices A and B are said to be equal, if
(i) order of A and B are same.
(ii) corresponding elements of A and B are same i.e. aij = bij, ∀ i and j.
e.g. [2013] and [2013] are equal matrices, but [3021] and [2031] are not equal matrices.

Operations on Matrices
Between two or more than two matrices, the following operations are defined below:
Addition and Subtraction of Matrices: Addition and subtraction of two matrices are defined in an order of both the matrices are same.
Addition of Matrix
If A = [aij]m×n and B = [yij]m×n, then A + B = [aij +bij]m×n, 1 ≤ i ≤ m, 1 ≤ j ≤ n
Subtraction of Matrix
If A = [aij]m×n and B = [bij]m×n, then A – B = [aij – bij]m×n, 1 ≤ i ≤ m, 1 ≤ j ≤ n

Properties of Addition of Matrices
(a) Commutative If A = [aij] and B = [bij] are matrices of the same order say m x n then A + B = B + A,
(b) Associative for any three matrices A = [aij], B = [bij], C = [cij] of the same order say m x n, A + (B + C) = (A + B) + C.
(c) Existence of additive identity Let A = [aij] be amxn matrix and O be amxn zero matrix, then A + O = O + A = A. In other words, O is the additive identity for matrix addition.
(d) Existence of additive inverse Let A = [aij]m×n be any matrix, then we have another matrix as -A = [-aij]m×n such that A + (-A) = (-A + A) = O. So, matrix (-A) is called additive inverse of A or negative of A.

Note
(i) If A and B are not of the same order, then A + B is not defined.
(ii) Addition of matrices is an example of a binary operation on the set of matrices of the same order.

Multiplication of a matrix by scalar number: Let A = [aij]m×n be a matrix and k is scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k, i.e. if A = [aij]m×n, then kA = [kaij]m×n.

Properties of Scalar Multiplication of a Matrix
Let A = [aij] and B = [bij]be two matrices of the same order say m × n, then
(a) k(A + B) = kA + kB, where k is a scalar.
(b) (k + l)A = kA + lA, where k and l are scalars.

Multiplication of Matrices: Let A and B be two matrices. Then, their product AB is defined, if the number of columns in matrix A is equal to the number of rows in matrix B.

Properties of Multiplication of Matrices
(a) Non-commutativity Matrix multiplication is not commutative i.e. if AB and BA are both defined, then it is not necessary that AB ≠ BA.
(b) Associative law For three matrices A, B, and C, if multiplication is defined, then A (BC) = (AB) C.
(c) Multiplicative identity For every square matrix A, there exists an identity matrix of the same order such that IA = AI = A.
Note: For Amxm, there is only one multiplicative identity Im.
(d) Distributive law For three matrices A, B, and C,
A(B + C) = AB + AC
(A + B)C = AC + BC
whenever both sides of the equality are defined.

Note: If A and B are two non-zero matrices, then their product may be  a zero matrix. 

Most important objective questions 🙋 :

Matrices Class 12 Maths MCQs Pdf

Question 1.
If A and B are symmetric matrices of the same order, then
(a) AB is a symmetric matrix
(b) A – Bis askew-symmetric matrix
(c) AB + BA is a symmetric matrix
(d) AB – BA is a symmetric matrix
Answer:
(c) AB + BA is a symmetric matrix

Question 2.
If A=[32x+3x−1x+2] is a symmetric matrix, then x =
(a) 4
(b) 3
(c) -4
(d) -3
Answer:
(c) -4

Question 3.
If A is a square matrix, then A – A’ is a
(a) diagonal matrix
(b) skew-symmetric matrix
(c) symmetric matrix
(d) none of these
Answer:
(b) skew-symmetric matrix

Question 4.
If A is any square matrix, then which of the following is skew-symmetric?
(a) A + AT
(b) A – AT
(c) AAT
(d) ATA
Answer:
(b) A – Question 5.

(a) α = a2 + b2, β = ab
(b) α = a2 + b2, β = 2ab
(c) α = a2 + b2, β = a2 – b2
(d) α = 2ab, β = a2 + b2
Answer:
(b) α = a2 + b2, β = 2ab

Question 6.
If A = ⎡⎣⎢100210x01⎤⎦⎥ and B = ⎡⎣⎢100−210y01⎤⎦⎥ and AB = I3, then x + y equals
(a) 0
(b) -1
(c) 2
(d) None of these
Answer:

(a) 0

Question 7.
If A = [1221] and f(x) = (1 + x) (1 – x), then f(a) is

Answer:
(a) −4[1111]

Question 8.
If A = [1334] and A2 – KA – 5I = 0, then K =
(a) 5
(b) 3
(c) 7
(d) None of these
Answer:
(a) 5

Question 9.

Answer:
(b) [−3−2107]

Question 10.
If matrix A = ⎡⎣⎢abcbcacab⎤⎦⎥ where a, b, c are real positivenumbers, abc = 1 and ATA = I, then the value of a3 + b3 + c3 is
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(d) 4

Question 11.

Answer:
(c) 111⎡⎣⎢−1−27−35−15−1−2⎤⎦⎥

Question 12.

Answer:
(a) −111,211

Question 13.
Using elementary transformation, find the inverse of matrix ⎡⎣⎢−113121231⎤⎦⎥

Answer:
(a) ⎡⎣⎢1−85−17−41−53⎤⎦⎥

Question 14.
Find the inverse of the matrix A=[1237], using elementary row transformation.

Answer:
(a) [7−2−31]

Question 15.

Answer:
(d) 12

Question 16.
Find the values of x, y, z respectively if the matrix A=⎡⎣⎢0xx2yy−yz−zz⎤⎦⎥ satisfy the equation ATA = I3.
(a) 12√⋅16√,13√
(b) −12√,−16√,−13√
(c) Both (a) and (b)
(d) None of these
Answer:
(c) Both (a) and (b)

Question 17.
If A=[cosxsinx−sinxcosx], find AAT.
(a) Zero Matrix
(b) I2
(c) [1111]
(d) None of these
Answer:
(b) I2

Question 18.
If A=⎡⎣⎢01−2−10−3230⎤⎦⎥, then A + 2AT equals
(a) A
(b) -AT
(c) AT
(d) 2A2
Answer:
(c) AT

Question 19.
For any square matrix A, AAT is a
(a) unit matrix
(b) symmetric matrix
(c) skew-symmetric matrix
(d) diagonal matrix
Answer:
(b) symmetric matrix

Question 20.
If A = ⎡⎣⎢649827531⎤⎦⎥ is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is

Answer:
(a) A=⎡⎣⎢667625751⎤⎦⎥

Question 21.
If the matrix A = ⎡⎣⎢5y422tx−3−7⎤⎦⎥ is a symmetric matrix, then find the value of x, y and t respectively.
(a) 4, 2, 3
(b) 4, 2, -3
(c) 4, 2, -7
(d) 2, 4, -7
Answer:
(b) 4, 2, -3

Question 22.
If a matrix A is both symmetric and skew-symmetric, then
(a) A is a diagonal matrix
(b) A is a zero matrix
(c) A is a scalar matrix
(d) A is a square matrix
Answer:
(b) A is a zero matrix

Question 23.
The matrix ⎡⎣⎢0−5750−11−7110⎤⎦⎥ is
(a) a skew-symmetric matrix
(b) a symmetric matrix
(c) a diagonal matrix
(d) an upper triangular matrix
Answer:
(a) a skew-symmetric matrix

Question 24.

Answer:
(a) 113⎡⎣⎢−1553−2−2−3152⎤⎦⎥

Question 25.

Answer:
(c) ⎡⎣⎢0−4−3−1331−2−2⎤⎦⎥

Question 26.

Answer:
(b) ⎡⎣⎢⎢⎢⎢01/31/23/5−1/301/51/3−1/2−1/501/7⎤⎦⎥⎥⎥⎥

Question 27.
The matrix A = [0110] is a
(a) unit matrix
(c) symmetric matrix
(b) diagonal matrix
(d) skew-symmetric matrix
Answer:
(d) skew-symmetric matrix

Question 28.
If [x+yx−y2x+z2z+w]=[40710], then the values of x, y, z and w respectively are
(a) 2, 2, 3, 4
(b) 2, 3, 1, 2
(c) 3, 3, 0, 1
(d) None of these
Answer:
(a) 2, 2, 3, 4

Question 29.

then find the values of a, b, c, x, y, and z respectively.
(a) -2, -7, -1, -3, -5, 2
(b) 2, 7, 1, 3, 5, -2
(c) 1, 3, 4, 2, 8, 9
(d) -1, 3, -2, -7, 4, 5
Answer:
(a) -2, -7, -1, -3, -5, 2

Question 30.
The order of the single matrix obtained from

(a) 2 × 3
(b) 2 × 2
(c) 3 × 2
(d) 3 × 3
Answer:
(d) 3 × 3

Question 31.
A=[12−1−1],B=[xy1−1] and (A + B)2 = A2 + B2, then x + y =
(a) 2
(b) 3
(c) 4
(d) 5
Answer:
(d) 5

Question 32.
If A2 – A + I = O, then the inverse of A is
(a) I – A
(b) A – I
(c) A
(d) A + I
Answer:
(a) I – A

Question 33.
Total number of possible matrices of order 3 × 3 with each entry 2 or 0 is
(a) 9
(b) 27
(c) 81
(d) 512
Answer:
(d) 512

Question 34.
The matrix ⎡⎣⎢05−8−50−128120⎤⎦⎥ is a
(a) diagonal matrix
(b) symmetric matrix
(c) skew symmetric matrix
(d) scalar matrix
Answer:
(c) skew symmetric matrix

Question 35.
If A is a matrix of order m × n and B is a matrix such that AB’ and B’A are both defined, then the order of matrix B is
(a) m × m
(b) n × n
(c) n × m
(d) m × n
Answer:
(d) m × n

Question 36.
If A and B are matrices of the same order, then (AB’ – BA’) is a
(a) skew-symmetric matrix
(b) null matrix
(c) symmetric matrix
(d) unit matrix
Answer:
(a) skew-symmetric matrix

Question 37.
If A is a square matrix such that A2 = I, then (A – I)3 + (A + I)3 – 7A is equal to
(a) A
(b) I – A
(c) I + A
(d) 3A
Answer:
(a) A

Question 38.
If A = ⎡⎣⎢211232112⎤⎦⎥, then A4 – 24 (A – I) =
(a) 5I + A
(b) 5I – A
(c) 5I
(d) 6I
Answer:
(b) 5I – A

Question 39.
If A is an m × n matrix such that AB and BA are both defined, then B is a
(a) m × n matrix
(b) n × m matrix
(c) n × n matrix
(d) m × n matrix
Answer:
(b) n × m matrix

Question 40.
If [1324], then A2 – 5A is equal to
(a) 2I
(b) 3I
(c) -2I
(d) null matrix
Answer:
(a) 2I

Question 41.

(a) A + B = B + A and A + (B + C) = (A + B) + C
(b) A + B = B + A and AC = BC
(c) A + B = B + A and AB = BC
(d) AC = BC and A = BC
Answer:
(a) A + B = B + A and A + (B + C) = (A + B) + C

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