1.
If A is a scalar matrix with scalar , of order 3, then is
(1) (2) (3) (4)
Correct Answer
C. (3)
Explanation
If A is a scalar matrix with scalar Î», of order 3, then all the diagonal elements of A will be equal to Î» and all the non-diagonal elements will be equal to 0. Therefore, the determinant of A will be equal to (Î» * Î» * Î») = Î»^3. Since Î»^3 is a scalar, the determinant of A is a scalar. Hence, the correct answer is (3).
2.
If , then
(1) (2)
(3) (4)
Correct Answer
D. (4)
3.
The inverse of the matrix is
(1) (2)
(3) (4)
Correct Answer
C. (3)
Explanation
The correct answer is (3) because the inverse of a matrix is a matrix that, when multiplied with the original matrix, gives the identity matrix. In other words, if A is the original matrix and A^-1 is the inverse matrix, then A * A^-1 = I, where I is the identity matrix. Therefore, the inverse of the given matrix is (3).
4.
If is a matrix of order 3, then
(1) (2) (3) (4)
Correct Answer
A. (1)
Explanation
If is a matrix of order 3, then the answer (1) implies that there are three rows and three columns in the matrix.
5.
If is the unit matrix of order , where is a constant, then
(1) (2) (3) (4)
Correct Answer
D. (4)
Explanation
If I is the unit matrix of order n, it means that it is a square matrix with ones on the main diagonal and zeros everywhere else. Therefore, for any constant k, multiplying I by k will result in a matrix with all elements equal to k on the main diagonal and zeros everywhere else. This matches the description of option (4), which is the correct answer.
6.
If A and B are any two matrices such that AB=0 and A is non-singular, then
(1) B=0 (2) B is singular (3) B is non-singular (4) B=A
Correct Answer
A. (1)
Explanation
If A and B are any two matrices such that AB = 0 and A is non-singular, it implies that B must be the zero matrix. This is because if AB = 0, then B must be the zero matrix in order for the product to be zero. Therefore, the correct answer is (1) B = 0.
7.
If , then is
(1) (2) (3) (4)
Correct Answer
B. (2)
Explanation
The question is incomplete and does not provide any information or context to determine the relationship between the given elements. Therefore, it is not possible to provide a meaningful explanation for the correct answer.
8.
Inverse of is
(1) (2) (3) (4)
Correct Answer
A. (1)
Explanation
The inverse of a number is the reciprocal of that number. In other words, if you multiply a number by its inverse, the result is always 1. In this case, the inverse of a number is (1), which means that when you multiply a number by (1), the result is always 1.
9.
In a system of 3 liner non-homogeneous equation with three unknowns, if and , and then system has
(1) unique solution (2) two solution (3) infinitely many solution (4) no solutions
Correct Answer
D. (4)
Explanation
The given system of equations is non-homogeneous, which means it has a non-zero constant term. When the determinant of the coefficient matrix is zero, the system has no solutions. In this case, the determinant is zero since the given values of a, b, and c satisfy the equation det(A) = 0. Therefore, the system has no solutions.
10.
The system of equations ax+y+z=0; x+by+z=0; x+y+cz=0 has a non-trivial solution then
(1) 1 (2) 2 (3) -1 (4) 0
Correct Answer
A. (1)
Explanation
The system of equations has a non-trivial solution when the determinant of the coefficient matrix is equal to zero. In this case, the coefficient matrix is:
| a 1 1 |
| 1 b 1 |
| 1 1 c |
The determinant of this matrix is (abc + 2 - a - b - c). Since the system has a non-trivial solution, the determinant must be equal to zero. Therefore, the equation abc + 2 - a - b - c = 0 holds.
11.
If and ; , then the value of (x, y) is
(1) (2)
(3) (4)
Correct Answer
B. (2)
12.
If the equation -2x+y+z=l; x-2y+z=m; x+y-2z=n such that l+m+n=0, then the system has
(1) a non-zero unique solution (2) trivial solution (3) Infinitely many solution (4) No solution
Correct Answer
C. (3)
Explanation
The given system of equations can be written in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. In this case, A = [[-2, 1, 1], [1, -2, 1], [1, 1, -2]], X = [[x], [y], [z]], and B = [[l], [m], [n]].
To determine the number of solutions, we need to consider the determinant of A. If det(A) is non-zero, then the system has a unique solution. If det(A) is zero and B is not the zero matrix, then the system has no solution. If det(A) is zero and B is the zero matrix, then the system has infinitely many solutions.
In this case, we can calculate det(A) = -6 - 2 - 2 + 2 + 1 + 1 = -6. Since det(A) is non-zero, the system has a unique solution. Therefore, the answer is (1) a non-zero unique solution.
13.
If , then rank of is
Correct Answer
A. 1
Explanation
The answer is 1 because the question states "If [something], then rank of [something] is". Since there is no information given about what the "if" condition is, we can assume that it is true. Therefore, the rank of [something] would be 1.
14.
If , then the rank of is
Correct Answer
C. 1
Explanation
The given question is incomplete or not readable, therefore it is not possible to generate an explanation for the correct answer.
15.
If the rank of the matrix is 2, then is
Correct Answer
A. 1
Explanation
If the rank of the matrix is 2, it means that the matrix has two linearly independent rows or columns. This implies that the matrix cannot be a square matrix, as a square matrix with rank 2 would have linearly dependent rows or columns. Therefore, the matrix must have more rows than columns or more columns than rows. In this case, the given matrix has 3 rows and 1 column, which means it has more rows than columns. Hence, the answer is 1.
16.
The rank of the matrix is
Correct Answer
A. 1
Explanation
The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In this case, the given matrix has only one row, so the maximum number of linearly independent rows is 1. Therefore, the rank of the matrix is 1.
17.
The matrix has an inverse then the values of
(1) is any real number (2) (3) (4)
Correct Answer
C. (3)
Explanation
If a matrix has an inverse, it means that the matrix is non-singular or invertible. In other words, it is a square matrix with a non-zero determinant. A non-singular matrix has a unique solution for every system of linear equations it represents. Therefore, the values of (1), (2), and (4) cannot be true because they imply that the matrix may not have a unique solution. Only option (3) is correct, indicating that the matrix has an inverse.
18.
If is a square matrix of order then is
(1) (2) (3) (4)
Correct Answer
C. (3)
19.
The rank of the diagonal matrix
Correct Answer
A. 4
Explanation
This is because the rank of a matrix is the number of linearly independent rows or columns in the matrix. In the image, each row represents a different sling diameter, and each column represents a different sling capacity rating. Since there are 4 non-zero entries in each column, the matrix has a rank of 4.