GATE 2021 AE – Question 47

Engineering Mathematics · Linear Algebra: Vector algebra, matrix algebra, systems of linear equations, rank, eigenvalues and eigenvectors · 2 marks · Numerical answer

The ratio of the product of eigenvalues to their sum for $A=\begin{bmatrix}3&1&2\\2&-3&-1\\1&2&1\end{bmatrix}$ is (nearest integer)

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Correct answer: 8

Explanation

For a square matrix, the **product of the eigenvalues is the determinant** and the **sum of the eigenvalues is the trace**.

$$A=\begin{bmatrix}3&1&2\\2&-3&-1\\1&2&1\end{bmatrix}$$

**Determinant** (expanding along the first row):
$$\det A=3\big((-3)(1)-(-1)(2)\big)-1\big((2)(1)-(-1)(1)\big)+2\big((2)(2)-(-3)(1)\big)$$
$$=3(-3+2)-1(2+1)+2(4+3)=-3-3+14=8 .$$

**Trace:** $3+(-3)+1=1$.

$$\frac{\text{product}}{\text{sum}}=\frac{8}{1}=\mathbf{8}.$$