CUET PG 2021 — Mathematics PYQ

Solution

If a square matrix $A$ of order $n$ has eigenvalues $a,b,c$, then the eigenvalues of its transpose matrix $A^T$ are exactly the same: $a,b,c$.

Proof/Reasoning

The eigenvalues of any matrix are the roots of its characteristic equation:

To find the eigenvalues of $A^T$, we look at its characteristic equation:

Since the identity matrix $I$ is symmetric ($I^T=I$), we can rewrite this as:

We know from the properties of determinants that the determinant of a matrix is equal to the determinant of its transpose, i.e., $\det (M)=\det (M^T)$. Therefore:

Since both matrices share the same characteristic polynomial, they must have the same eigenvalues.

Summary:

The eigenvalues of $A^T$ are $\{a,b,c\}$.