Question

Following are the statements for a square matrix A:
A. If A is invertible then $det(A) \neq 0$
B. If $det(A) = 0$, A has no inverse
C. All eigen values of A are always positive
D. $A^T A$ is always invertible if A is invertible
Choose the correct answer from the options given below :

• A. A, B and C only
• B. A and B only
• C. A, B and D only
• D. A, B, C and D

Hint:

Remember: $det(A) = \text{Product of Eigenvalues}$. If any eigenvalue is zero, the determinant is zero, and the matrix is not invertible.

Answer 1

The Correct Option is C

Concept: Matrix invertibility is fundamentally linked to the determinant. A matrix is invertible (non-singular) if and only if its determinant is non-zero.

Step 1: Evaluate statements A and B.
A square matrix $A$ is invertible if there exists a matrix $B$ such that $AB = I$. This is true if and only if $|A| \neq 0$. Thus, statement (A) is correct. Conversely, if $|A| = 0$, the matrix is singular and cannot be inverted. Thus, statement (B) is also correct.

Step 2: Evaluate statement C.
Eigenvalues of a matrix can be positive, negative, zero, or even complex. For example, a matrix with a negative determinant will have at least one negative eigenvalue. Only positive definite matrices have strictly positive eigenvalues. Thus, (C) is false.

Step 3: Evaluate statement D.
If $A$ is invertible, then $det(A) \neq 0$. Since $det(A^T A) = det(A^T) \cdot det(A) = (det(A))^2$, and $(det(A))^2$ must be positive (and non-zero), $A^T A$ is always invertible. Thus, (D) is correct.