1
What is the order of the matrix ?
Correct Answer:
Explanation: The order of a matrix is defined as , where is the number of rows and is the number of columns. Matrix has 2 rows and 3 columns.
2
A matrix in which the number of rows is equal to the number of columns is called a:
A. Row matrix
B. Column matrix
C. Square matrix
D. Rectangular matrix
Correct Answer: Square matrix
Explanation: A square matrix is a matrix with the same number of rows and columns ().
3
If is a matrix of order , then is a row matrix if:
Correct Answer:
Explanation: A row matrix has only one row, so must equal 1.
4
Given and , what is ?
Correct Answer:
Explanation: Matrix addition is performed element-wise: , , , .
5
Two matrices and can be added only if:
A. They are both square matrices
B. They have different orders
C. They have the same order
D. The number of columns of equals the number of rows of
Correct Answer: They have the same order
Explanation: Matrix addition and subtraction are only defined for matrices of the exact same dimensions ().
6
If , what is ?
Correct Answer:
Explanation: In scalar multiplication, every element of the matrix is multiplied by the scalar. , , etc.
7
If matrix is of order and matrix is of order , then the order of matrix is:
Correct Answer:
Explanation: The resulting matrix from multiplication has the row count of the first matrix and the column count of the second matrix.
8
Matrix multiplication is generally:
A. Commutative ()
B. Not Commutative ()
C. Always equal to the Identity matrix
D. Always equal to the Zero matrix
Correct Answer: Not Commutative ()
Explanation: In general, for matrices, even if both products are defined.
9
If , then is known as a:
A. Zero Matrix
B. Identity Matrix
C. Rectangular Matrix
D. Singular Matrix
Correct Answer: Identity Matrix
Explanation: A square matrix with 1s on the main diagonal and 0s elsewhere is an Identity Matrix (denoted as ).
10
What is the transpose of ?
Correct Answer:
Explanation: The transpose () is formed by swapping rows and columns. The first column of becomes the first row of .
11
For any matrix , is equal to:
Correct Answer:
Explanation: Transposing a matrix twice returns the matrix to its original configuration.
12
If is a square matrix such that , then is called a:
A. Skew-symmetric matrix
B. Symmetric matrix
C. Diagonal matrix
D. Scalar matrix
Correct Answer: Symmetric matrix
Explanation: A symmetric matrix is equal to its own transpose.
13
Which of the following properties is true for the transpose of a product of matrices?
Correct Answer:
Explanation: The transpose of a product is the product of the transposes in reverse order (Reversal Law of Transposes).
14
Calculate the determinant of :
Correct Answer: 6
Explanation: For a matrix, the determinant is . Here, .
15
A matrix is said to be singular if its determinant is:
A. 1
B. -1
C.
D. Undefined
Correct Answer:
Explanation: A square matrix is singular if and only if its determinant is zero.
16
Find the value of if the matrix is singular.
Correct Answer: 4
Explanation: For a singular matrix, the determinant is 0. .
17
If two rows (or columns) of a determinant are identical, the value of the determinant is:
A. 1
B. Does not change
C.
D. Double the original value
Correct Answer:
Explanation: This is a fundamental property of determinants. If two parallel lines (rows or columns) are identical, the determinant vanishes.
18
If , what is the trace of ?
Correct Answer: 5
Explanation: The trace of a matrix is the sum of the elements on the principal diagonal: .
19
Calculate the determinant of :
Correct Answer: 6
Explanation: The determinant of a diagonal matrix is the product of its diagonal elements: .
20
The cofactor of an element denoted by is related to the minor by the formula:
Correct Answer:
Explanation: The cofactor is the signed minor. The sign depends on the sum of the row and column indices.
21
The Adjoint of a matrix , denoted as , is defined as:
A. The matrix of cofactors
B. The transpose of the matrix of cofactors
C. The inverse of the matrix of minors
D. The transpose of the matrix itself
Correct Answer: The transpose of the matrix of cofactors
Explanation: To find the adjoint, you form the matrix of cofactors and then take its transpose.
22
The inverse of a square matrix exists if and only if:
A.
B.
C. is an identity matrix
D. is a zero matrix
Correct Answer:
Explanation: A matrix must be non-singular () to be invertible.
23
The formula for the inverse of a matrix is:
Correct Answer:
Explanation: The inverse is calculated by dividing the adjoint matrix by the determinant of the matrix.
24
Find the inverse of :
Correct Answer:
Explanation: . For a , swap diagonal elements () and change signs of off-diagonals. Then divide by .
25
If and are invertible matrices of the same order, then is equal to:
Correct Answer:
Explanation: This is the Reversal Law for Inverses: the inverse of a product is the product of the inverses in reverse order.
26
If and , calculate .
Correct Answer:
Explanation: Row 1 Col 1: . Row 1 Col 2: . Row 2 Col 1: . Row 2 Col 2: .
27
Which of the following is true regarding the determinant of a transpose?
Correct Answer:
Explanation: The value of a determinant remains unchanged if its rows and columns are interchanged.
28
If every element in a row (or column) of a square matrix is zero, then the determinant is:
A. 1
B. The product of diagonal elements
C.
D. Infinity
Correct Answer:
Explanation: If a row or column consists entirely of zeros, the determinant is zero.
29
Calculate the minor for the matrix .
Correct Answer: -6
Explanation: Remove row 1 and column 2. Left with . Calc: .
30
If is a matrix of order and , what is the value of ?
Correct Answer: 32
Explanation: For an matrix, . Here, . So, .
31
A matrix is called orthogonal if:
Correct Answer:
Explanation: An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors, meaning .
32
What is the result of ?
Correct Answer:
Explanation: The product of a matrix and its adjoint is a scalar matrix where the diagonal elements equal the determinant of .
33
If and are matrices of order , then is equal to:
Correct Answer:
Explanation: The determinant of a product is the product of the determinants.
34
In the adjoint method for finding the inverse, if , what is ?
Correct Answer:
Explanation: For a matrix, swap the diagonal elements and change the signs of the off-diagonal elements.
35
If , then is:
A. Symmetric
B. Skew-symmetric
C. Identity
D. Zero Matrix
Correct Answer: Skew-symmetric
Explanation: For any square matrix , is always a skew-symmetric matrix.
36
The inverse of the Identity matrix is:
Correct Answer:
Explanation: , so the inverse of is itself.
37
Which of the following operations changes the value of a determinant?
A. Swapping two rows
B. Adding a multiple of one row to another
C. Transposing the matrix
D. Replacing a row with linear combination of other rows
Correct Answer: Swapping two rows
Explanation: Swapping two rows (or columns) changes the sign of the determinant.
38
If , then is a:
A. Scalar Matrix
B. Unit Matrix
C. Row Matrix
D. Zero Matrix
Correct Answer: Scalar Matrix
Explanation: A diagonal matrix where all diagonal elements are equal is called a Scalar Matrix.
39
Given , calculate .
Correct Answer:
Explanation: . Row 1 Col 1: . Row 1 Col 2: . Row 2 Col 1: . Row 2 Col 2: .
40
If is a skew-symmetric matrix, then equals:
Correct Answer:
Explanation: By definition, a matrix is skew-symmetric if .
41
Evaluate the determinant:
Correct Answer: 2
Explanation: Using row operations () simplifies to . Expand along : .
42
If and are matrices, then is equal to:
Correct Answer:
Explanation: The transpose of a sum is the sum of the transposes.
43
A matrix such that is called:
A. Idempotent matrix
B. Nilpotent matrix
C. Involutory matrix
D. Orthogonal matrix
Correct Answer: Idempotent matrix
Explanation: An idempotent matrix is a matrix which, when multiplied by itself, yields itself.
44
If and , find .
Correct Answer:
Explanation: is , is . Result is . .
45
If and , find .
Correct Answer:
Explanation: is , is . Result is . R1C1: , R1C2: , etc.
46
For a non-singular matrix , is same as:
Correct Answer:
Explanation: Transpose and Inverse operations are commutative: the transpose of the inverse is the inverse of the transpose.
47
The diagonal elements of a skew-symmetric matrix are always:
A. 1
B. Equal
C.
D. Non-zero
Correct Answer:
Explanation: For , .
48
Which of the following is NOT a property of matrix addition?
Correct Answer:
Explanation: Matrix addition is commutative and associative, and has an identity (0), but it does not equal the product of the matrices.
49
If , the cofactor is:
Correct Answer: -6
Explanation: Remove row 2, col 1. The minor is 6. The position is , so sign is . Result is .
50
What is the determinant of a Identity matrix?
A.
B. 1
C. 3
D. Undefined
Correct Answer: 1
Explanation: The determinant of any Identity matrix is 1.