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GATE 2016 EC (EC2) – Question 39

Engineering Mathematics · Linear Algebra · 2 marks · Numerical answer

The matrix
$$A=\begin{bmatrix}a&0&3&7\\2&5&1&3\\0&0&2&4\\0&0&0&b\end{bmatrix}$$
has $\det A=100$ and $\operatorname{tr}A=14$. The value of $|a-b|$ is ________.

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Correct answer: 2.9 to 3.1

Explanation

The matrix is block upper triangular. Its determinant is $(5a)(2b)=10ab$, so ab=10. Its trace is a+5+2+b=14, so a+b=7.

Consequently $(a-b)^2=(a+b)^2-4ab=49-40=9$. Taking the positive square root gives **$|a-b|=3$**. The two values are 2 and 5 in either order.

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