Match List-I with List-IIList-1 List-II (A) $A^T = A$ (I) A is a singular matrix (B) $A^T = -A$ (II) A is a non-singular matrix (C) $|A| = 0$ (III) A is a skew symmetric matric (D) $|A| \neq 0$ (IV) A is a symmetric matric
Choose the correct answer from the options given below:
Matrix Properties Explained
This question tests the understanding of fundamental matrix properties related to the transpose of a matrix ($A^T$) and its determinant ($|A|$).
Understanding Matrix Types
- Symmetric Matrix: A square matrix $A$ is symmetric if its transpose is equal to the matrix itself. Mathematically, this is represented as $A^T = A$. The elements across the main diagonal are mirrored.
- Skew-Symmetric Matrix: A square matrix $A$ is skew-symmetric if its transpose is equal to the negative of the matrix. Mathematically, this is represented as $A^T = -A$. For a skew-symmetric matrix, the diagonal elements must be zero.
- Singular Matrix: A square matrix $A$ is singular if its determinant is zero. Mathematically, this is $|A| = 0$. Singular matrices do not have an inverse.
- Non-singular Matrix: A square matrix $A$ is non-singular if its determinant is not zero. Mathematically, this is $|A| \neq 0$. Non-singular matrices have a unique inverse.
Matching Matrix Conditions to Definitions
We need to match the conditions given in List-I with their corresponding definitions or properties in List-II.
- (A) $A^T = A$: This condition defines a symmetric matrix. Therefore, (A) matches with (IV).
- (B) $A^T = -A$: This condition defines a skew-symmetric matrix. Therefore, (B) matches with (III).
- (C) $|A| = 0$: This condition signifies that the matrix $A$ has a determinant of zero, which means it is a singular matrix. Therefore, (C) matches with (I).
- (D) $|A| \neq 0$: This condition signifies that the matrix $A$ has a non-zero determinant, which means it is a non-singular matrix. Therefore, (D) matches with (II).
Identifying the Correct Option
Based on the matching derived above, the correct pairings are:
- (A) - (IV)
- (B) - (III)
- (C) - (I)
- (D) - (II)
This set of matches corresponds to Option 2.
Important Questions from Matrices
The eigenvalues of the 3 × 3 matrix M = \(\left(\begin{array}{lll}\rm a^2 & \rm a b & \rm a c \\ \rm a b & \rm b^2 & \rm b c \\ \rm a c & \rm b c &\rm c^2\end{array}\right)\) are
A generic 3 × 3 real matrix A has eigenvalues 0, 1 and 6, and I is the 3 × 3 identity matrix. The quantity/quantities that cannot be determined from this information is/are the
If \(A = \(\left[ {\begin{array}{} {coshx}&{sinhx}\\ { - sinhx}&{coshx} \end{array}} \right])\) , then trace (A 2) is equal to
Let A be a non-singular diagonalisable matrix of order 3 with eignvalues λ1, λ2, λ3. A -1 is diagonalisable if:
Assertion(A): If A is any Matrix given by A = \(\left(\begin{array}{ccc}5 & 0 & 3 \\ −1 & 0 & 2 \\ 1 & 0 & 1\end{array}\right)\)
Then det A = 0, since all elements in column II are zero
Reason (R): Laplace expansion permits evaluation of a determinant along any row or column
