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.
Matrix Multiplication Rules Explained
To determine if a matrix product is defined, we look at the dimensions (order) of the matrices involved. If matrix A has dimensions \(m \times n\) and matrix B has dimensions \(p \times q\), their product AB is defined only if the number of columns in A (n) is equal to the number of rows in B (p). That is, \(n = p\). The resulting matrix AB will have dimensions \(m \times q\).
In this problem, we are given:
- Matrix X has order \(3 \times 3\).
- Matrix Y has order \(2 \times 3\).
- Matrix Z has order \(3 \times 2\).
Let's analyze each statement based on these rules.
Statement I: (ZY)X is defined
First, consider the product ZY.
- Z has order \(3 \times 2\).
- Y has order \(2 \times 3\).
Since the number of columns in Z (2) equals the number of rows in Y (2), the product ZY is defined. The resulting matrix ZY has order \(3 \times 3\).
Next, consider the product (ZY)X.
- ZY has order \(3 \times 3\).
- X has order \(3 \times 3\).
Since the number of columns in ZY (3) equals the number of rows in X (3), the product (ZY)X is defined. The resulting matrix (ZY)X has order \(3 \times 3\). A \(3 \times 3\) matrix is a square matrix of order 3.
Therefore, statement I is correct.
Statement II: Y(XZ) is defined
First, consider the product XZ.
- X has order \(3 \times 3\).
- Z has order \(3 \times 2\).
Since the number of columns in X (3) equals the number of rows in Z (3), the product XZ is defined. The resulting matrix XZ has order \(3 \times 2\).
Next, consider the product Y(XZ).
- Y has order \(2 \times 3\).
- XZ has order \(3 \times 2\).
Since the number of columns in Y (3) equals the number of rows in XZ (3), the product Y(XZ) is defined. The resulting matrix Y(XZ) has order \(2 \times 2\). A \(2 \times 2\) matrix is a square matrix of order 2.
Therefore, statement II is correct.
Statement III: X(YZ) is not defined
First, consider the product YZ.
- Y has order \(2 \times 3\).
- Z has order \(3 \times 2\).
Since the number of columns in Y (3) equals the number of rows in Z (3), the product YZ is defined. The resulting matrix YZ has order \(2 \times 2\).
Next, consider the product X(YZ).
- X has order \(3 \times 3\).
- YZ has order \(2 \times 2\).
The number of columns in X is 3, and the number of rows in YZ is 2. Since \(3 \neq 2\), the product X(YZ) is not defined.
Therefore, statement III is correct.
Conclusion on Statements
Based on the analysis:
- Statement I is correct.
- Statement II is correct.
- Statement III is correct.
Since all three statements are correct, the option including I, II, and III is the correct choice.
Similar Questions
- If \(M_k = \begin{bmatrix} k & k-1 \\ k-1 & k \end{bmatrix}\) where k is a natural number, then what is \(|M_1| + |M_2| + |M_3| + \dots + |M_{50}|\) equal to ?
- Let \(f(x) = \begin{vmatrix} 3x^2 & \cos x & -\sin x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}\) where q is any constant, then what is \(\frac{d^2}{dx^2}(f(x))\) at \(x = 0\) equal to ?
- If p, q, r are the cube roots of unity, then what is \(\begin{vmatrix} p^2+q^2 & r^2 & r^2 \\ p^2 & q^2+r^2 & p^2 \\ q^2 & q^2 & r^2+p^2 \end{vmatrix}\) equal to ?
- If A is a square matrix such that \(|A| = -2\), then \(|AA^T|\), where \(A^T\) is the transpose of A, is equal to
- Consider the following statements :
I. If \(n \times n\) (\(n > 1\)) matrix is symmetric, then its inverse is also a symmetric matrix.
II. If \(n \times n\) (\(n > 1\)) matrix is singular, then its adjoint is also a singular matrix.
Which of the statements given above is/are correct ? - If \(M = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}\), then what is the value of \(|M| |adjM|\) ?
- If \(M^T\) is the transpose of a \(2 \times 2\) matrix M, then which of the following is/are correct ?
I. \(|M + M^T| = |M| + |M^T|\) if M is symmetric.
II. \(|M + M^T| = 0\) if M is anti-symmetric.
Select the answer using the code given below : - If M is a square matrix such that \(M^3 = M\), then how many values of \(|M|\) are possible ?
- Let \(p = (x + y + z)\) and \(q = xyz\). If \(\begin{vmatrix} x & 1 & 1 \\ 1 & y & 1 \\ 1 & 1 & z \end{vmatrix}\) is positive, then which one of the following is correct ?
- Let A and B be two square matrices of same order. If AB is a null matrix, then which one of the following is correct?
Important Questions from Matrices and Determinants
- If each element of a matrix is zero, then the matrix is:
- If \(M_k = \begin{bmatrix} k & k-1 \\ k-1 & k \end{bmatrix}\) where k is a natural number, then what is \(|M_1| + |M_2| + |M_3| + \dots + |M_{50}|\) equal to ?
- Let \(f(x) = \begin{vmatrix} 3x^2 & \cos x & -\sin x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}\) where q is any constant, then what is \(\frac{d^2}{dx^2}(f(x))\) at \(x = 0\) equal to ?
- If p, q, r are the cube roots of unity, then what is \(\begin{vmatrix} p^2+q^2 & r^2 & r^2 \\ p^2 & q^2+r^2 & p^2 \\ q^2 & q^2 & r^2+p^2 \end{vmatrix}\) equal to ?
- If A is a square matrix such that \(|A| = -2\), then \(|AA^T|\), where \(A^T\) is the transpose of A, is equal to
