Q.If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, show that $A^2 - 5A + 7I = 0$.

We verify the relation by computing $A^2$ directly and substituting; no theorem beyond matrix multiplication is required. For $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, …

We must show $A^2 - 5A + 7I = O$ for $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, where $I$ is the $2\times2$ identity and $O$ the zero matrix. The safest in-syllabus method is to compute $A^2$ by direct multiplication and substitute.

Step 1 — Compute $A^2$

Step 2 — Compute $5A$ and $7I$

Step 3 — Substitute …

Method: Verifying a matrix satisfies a polynomial via its characteristic equation

Use this whenever you must show a square matrix $A$ satisfies an equation like $A^2 + pA + qI = O$ (a $2\times2$ Cayley–Hamilton verification).

Steps

Step 1: Build the characteristic polynomial.

For any $2\times2$ matrix, it is

where $\operatorname{tr}A$ is the sum of the diagonal entries and $\det A = ad - bc$.

Step 2: Invoke Cayley–Hamilton. …

Common Mistakes

Mistake 1: Treating the constant term as a scalar instead of $qI$.

Why it's wrong: you cannot add a plain number to a matrix; the "$+7$" must be $7I$. Correct approach: replace every constant in the polynomial by that constant times the identity matrix.

Mistake 2: Getting the sign of the trace or determinant term wrong. …