Matrices Problem

Algebra Level 2

If matrix $A=\left(\begin{array}{cc}a& b\\b& a \end{array}\right)$ satisfies

$A^2-10A+16I=0,$

where $b>0$ and $I=\left(\begin{array}{cc}1&0\\0&1 \end{array}\right),$ what is the value of $a-b?$

1 2 -1 0

2 solutions

Victor Porto
Oct 9, 2014

(I) a² + b² - 10a + 16 = 0

(II) 2ab - 10b = 0

(I) 2a - 10 = 0

a = 5

(II)25 + b² - 50 + 16 = 0

-9 + b² = 0

b = 3

a - b

5 - 3

Mudit Jha
Jul 7, 2014

Write RHS as 2 X 2 matrix with each element as 0 (stupidly obvious but helps to compare different elements as the other matrices are 2 X 2 as well). Use matrix multiplication and scalar multiplication on the identity matrix.

Matrices Problem

2 solutions

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