Consider the matrix A - - 2 1 1- 2 3 4- -1 -1 -2 - whose eigen values are 1- -1 and 3- The trace A3 - 3A2 is

∵ A has the eigen values 1, 1, 3. ⇒ (A^3 3A^2) has the eigen values [( 1)^3 3( 1)^2], [(1)^3 3(1)^2] and [(3)^3 3(3)^2] = 4, 2, 0 So, Trace( A^3 3A^2 ) ...

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Engineering Mathematics

Properties III

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Consider the matrix A = $⎡ ⎢ ⎣ \begin{matrix} 2 & 1 & 1 2 & 3 & 4 - 1 & - 1 & - 2 \end{matrix} ⎤ ⎥ ⎦$ whose eigen values are 1, -1 and 3. The trace $(A^{3} - 3 A^{2})$ is
-6

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Solution

The correct option is A -6
$∵$ A has the eigen values -1, 1, 3.

$\Rightarrow$ $(A^{3} - 3 A^{2})$ has the eigen values $[(- 1)^{3} - 3 (- 1)^{2}]$ , $[(1)^{3} - 3 (1)^{2}]$ and $[(3)^{3} - 3 (3)^{2}]$ = -4, -2, 0

So, Trace( $A^{3} - 3 A^{2}$ )= Sum of eigen values
= $- 4 - 2 + 0 = - 6$

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Consider the matrix A = ⎡⎢⎣211234−1−1−2⎤⎥⎦ whose eigen values are 1, -1 and 3. The trace (A3−3A2) is -6

The correct option is A -6∵ A has the eigen values -1, 1, 3. ⇒ (A3−3A2) has the eigen values [(−1)3−3(−1)2], [(1)3−3(1)2] and [(3)3−3(3)2] = -4, -2, 0 So, Trace( A3−3A2 )= Sum of eigen values =−4−2+0=−6

Consider the matrix A = $⎡ ⎢ ⎣ \begin{matrix} 2 & 1 & 1 2 & 3 & 4 - 1 & - 1 & - 2 \end{matrix} ⎤ ⎥ ⎦$ whose eigen values are 1, -1 and 3. The trace $(A^{3} - 3 A^{2})$ is
-6

The correct option is A -6
$∵$ A has the eigen values -1, 1, 3.

$\Rightarrow$ $(A^{3} - 3 A^{2})$ has the eigen values $[(- 1)^{3} - 3 (- 1)^{2}]$ , $[(1)^{3} - 3 (1)^{2}]$ and $[(3)^{3} - 3 (3)^{2}]$ = -4, -2, 0

So, Trace( $A^{3} - 3 A^{2}$ )= Sum of eigen values
= $- 4 - 2 + 0 = - 6$