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. Square matrix
C. Column 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 have the same order
B. The number of columns of equals the number of rows of
C. They have different orders
D. They are both square matrices
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. Not Commutative ()
B. Always equal to the Identity matrix
C. Always equal to the Zero matrix
D. Commutative ()
Correct Answer: Not Commutative ()
Explanation:
In general, for matrices, even if both products are defined.
9
If , then is known as a:
A. Rectangular Matrix
B. Singular Matrix
C. Identity Matrix
D. Zero 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. Diagonal matrix
B. Skew-symmetric matrix
C. Scalar matrix
D. Symmetric 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. 0
C. 1
D. Undefined
Correct Answer: 0
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. Does not change
B. 0
C. Double the original value
D. 1
Correct Answer: 0
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 inverse of the matrix of minors
B. The transpose of the matrix of cofactors
C. The transpose of the matrix itself
D. The matrix of cofactors
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. Infinity
B. 0
C. The product of diagonal elements
D. 1
Correct Answer: 0
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. Zero Matrix
C. Identity
D. Skew-symmetric
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. Transposing the matrix
B. Adding a multiple of one row to another
C. Swapping two rows
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. Unit Matrix
B. Zero Matrix
C. Row Matrix
D. Scalar 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. Orthogonal matrix
B. Idempotent matrix
C. Involutory matrix
D. Nilpotent matrix
Correct Answer: Idempotent matrix
Explanation:
An idempotent matrix is a matrix which, when multiplied by itself, yields itself.
44
If and , find .
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. Non-zero
B. Equal
C. 0
D. 1
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. Undefined
B. 3
C. 1
D. 0
Correct Answer: 1
Explanation:
The determinant of any Identity matrix is 1.