SENAPREP
Types of matrices Questions
Chapter: Matrices and Determinants | Total Available: 19 Real Questions
How many matrices of different orders are possible with elements comprising all prime numbers less than 30?
Explanation:
Prime numbers less than 30 are 2, 3, 5, 7, 11, 13,
17, 19, 23 and 29.
So, there are 10 prime numbers less than 30.
Now, different orders possible including all ten
prime numbers are 1 × 10, 10 × 1, 2 × 5, 5 × 2.
? 4 matrices of different orders are possible
with elements comprising all prime numbers
less than 30.
Hint:
• There are 10 prime numbers less than 30.
• Number of elements in matrix of order
m × n are mn.
If A and B are two matrices such that AB is of order n × n, then which one of the following is correct?
If M is a square matrix such that
M³ = M,
then how many values of |M| are possible?
Solution
Option (c) is correct.
Explanation:
Given the condition \( M^3 = M \), we can factor it as follows: \[ M^3 - M = 0 \] This can be factored into: \[ M(M^2 - I) = 0 \] where \( I \) is the identity matrix. Thus, we have two cases: either \( M = 0 \) or \( M^2 = I \). The eigenvalues of \( M \), denoted by \( \lambda \), must satisfy the equation \( \lambda^3 - \lambda = 0 \). Factoring gives: \[ \lambda(\lambda^2 - 1) = 0 \implies \lambda(\lambda - 1)(\lambda + 1) = 0 \] This yields the eigenvalues \( \lambda = 0, 1, -1 \). The determinant of \( M \) is the product of its eigenvalues. Possible combinations of eigenvalues for a \( n \times n \) matrix yield different determinants: - If all eigenvalues are \( 0 \): \( \text{det}(M) = 0 \) - If all eigenvalues are \( 1 \): \( \text{det}(M) = 1 \) - If one eigenvalue is \( 0 \) and others can be \( 1 \) or \( -1\): - \( \text{det}(M) = 0 \) - \( \text{det}(M) = 1 \) - \( \text{det}(M) = -1 \) Thus, the possible values of \( |M| \) are \( 0, 1, -1 \). Therefore, there are three unique values possible for \( |M| \).- Cubic equation and its roots.
- Eigenvalue properties of matrices.
A flowchart outlining the relationship between \( M^3 \), its factored forms, eigenvalues, and how they relate to determining the determinant.
Verified against NCERT Class XI Mathematics, Chapter 5 (Linear Algebra).
If A=
Where x, y and z are integers, is an orthogonal matrix, then what is the value of x2 + y2 + z2?
Let X be a matrix of order 3 × 3, Y be a matrix of order 2 × 3 and Z be a matrix of order 3 × 2. Which of the following statements are correct?
(I) (ZY)X is defined and is a square matrix of order 3. (II) Y(XZ) is defined and is a square matrix of order 2.
(III) X(YZ) is not defined. Select the answer using the code given below.
What is the number of different matrices, each having 4 entries that can be formed using 1, 2, 3, 4 (repetition is allowed)?
Possible order of 4 entries matrices
1 × 4, 2 × 2, 4 × 1
Total number of matrices = 4 × 4 × 4 × 4 × 3 =
768
Consider the following statements in respect of a skew-symmetric matrix A of order 3:
1. All diagonal elements are zero.
2. The sum of all the diagonal elements of the matrix is zero.
3. A is orthogonal matrix.
Which of the statements given above are correct?
We know that all diagonal elements of skew symmetric matrix is zero then their sum is also zero. So, statements 1 and 2 are correct. We know that If
AAT = I then A is called orthogonal matrix.
But AAT = A(–A) = –A2 [? AT = –A]
So, Statement 3 is wrong. SOLVED PAPER – 2024-I
Consider the following statements in respect of non-singular matrix A of order n:
1. A(adj AT) = A(adj A)T
2. If A2 = A, then A is identity matrix of order n
3. If A3 = A, then A is identity matrix of order n
Which of the statements given above are correct?
By properties statement 1 is correct. We know that If An = A then A is identity matrix. \ Statements 2 and 3 are correct. So, all statements are correct.
1. A" will always be singular for any positive integer n.
2. A" will always be a diagonal matrix for anypositive integer n.
3. A" will always be a symmetric matrix for any positive integer n.
Select the correct answer using the code given below:
