JEE MainMathematicsEasyMatrix and determinants

If $A^{3} = \lbrack\begin{matrix}
8 & - 57 \\
0 & 27
\end{matrix}\rbrack$, where A is an upper triangular matrix with real
entries, then $det(5A - A^{2})$ is equal to

Show answer and solution

Let $A = \lbrack\begin{matrix}
a & b \\
0 & c
\end{matrix}\rbrack,A^{2} = \lbrack\begin{matrix}
a^{2} & ab + bc \\
0 & c^{2}
\end{matrix}\rbrack$
$A^{3} = \lbrack\begin{matrix}
a^{3} & a^{2}b + abc + bc^{2} \\
0 & c^{3}
\end{matrix}\rbrack$
$A^{3} = \lbrack\begin{matrix}
8 & - 57 \\
0 & 27
\end{matrix}\rbrack$

a^{3} = 8{\ \ \ \ \ \ \ \ \ \ \ \ c}^{3} = 27\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a^{2}b + abc + bc^{2} = - 57

a = 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ c = 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ b = - 3

Source: Grass-G / M-17 Mathematics Paper 26 Nov Second.docx

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If $A^{3} = \lbrack\begin{matrix}8 & - 57 \\0 & 27\end{matrix}\rbrack$, where A is an upper triangular matrix with realentries, then det(5A−A2)det(5A - A^{2})det(5A−A2) is equal to

Need a video for this idea?

If $A^{3} = \lbrack\begin{matrix}
8 & - 57 \\
0 & 27
\end{matrix}\rbrack$, where A is an upper triangular matrix with real
entries, then $det(5A - A^{2})$ is equal to