Name: Understanding Trace, Transpose, and Conjugate of Matrix
Uploaded: 2019-07-19
Duration: 20 min 56 s
Description: Trace, transpose, and conjugate are the three important terms often used with respect to matrices. In this video, we will go through these terminologies and...

Question

If

A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{matrix}]

,and

I

is an identity matrix of order

3

, show that

A^{3} = p I + q A + r A^{2}

.

Accepted Answer

Given, $A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{matrix}]$

Now, $A^{2} = A \times A$

$\Rightarrow A^{2} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{matrix}] [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{matrix}]$

$\Rightarrow A^{2} = [\begin{matrix} 0 + 0 + 0 & 0 + 0 + 0 & 0 + 1 + 0 \\ 0 + 0 + p & 0 + 0 + q & 0 + 0 + r \\ 0 + 0 + r p & p + 0 + r q & 0 + q + r^{2} \end{matrix}]$

$A^{2} = [\begin{matrix} 0 & 0 & 1 \\ p & q & r \\ r p & p + r q & q + r^{2} \end{matrix}]$

Now $A^{3} = A^{2} \times A$

$\Rightarrow A^{3} = [\begin{matrix} 0 & 0 & 1 \\ p & q & r \\ r p & p + r q & q + r^{2} \end{matrix}] [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{matrix}]$

$\Rightarrow A^{3} = [\begin{matrix} 0 + 0 + p & 0 + 0 + q & 0 + 0 + r \\ 0 + 0 + r p & p + 0 + r q & 0 + q + r^{2} \\ 0 + 0 + p q + r^{2} p & r p + 0 + q^{2} + r^{2} q & 0 + p + r q + r q + r^{3} \end{matrix}]$

$\Rightarrow A^{3} = [\begin{matrix} p & q & r \\ r p & p + r q & q + r^{2} \\ p q + r^{2} p & r p + q^{2} + r^{2} q & p + 2 r q + r^{3} \end{matrix}] . . . (1)$

Now let us find $p I + q A + r A^{2}$

$\Rightarrow p I + q A + r A^{2} = p [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] + q [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{matrix}] + r [\begin{matrix} 0 & 0 & 1 \\ p & q & r \\ r p & p + r q & q + r^{2} \end{matrix}]$

$\Rightarrow p I + q A + r A^{2} = [\begin{matrix} p & 0 & 0 \\ 0 & p & 0 \\ 0 & 0 & p \end{matrix}] + [\begin{matrix} 0 & q & 0 \\ 0 & 0 & q \\ p q & q^{2} & q r \end{matrix}] + [\begin{matrix} 0 & 0 & r \\ r p & r q & r^{2} \\ r^{2} p & r p + r^{2} q & r q + r^{3} \end{matrix}]$

$\Rightarrow p I + q A + r A^{2} = [\begin{matrix} p & q & r \\ r p & p + r q & q + r^{2} \\ p q + r^{2} p & q^{2} + r^{2} q + r p & p + 2 q r + r^{3} \end{matrix}] . . . (2)$

Therefore, $A^{3} = p I + q A + r A^{2}$ [From $(1)$ and $(2)$ ]

If $A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{matrix}]$ ,and $I$ is an identity matrix of order $3$ , show that $A^{3} = p I + q A + r A^{2}$ .

Important Questions on Algebra of Matrices

Learn and Prepare for any exam you want !

If A=010001pqr ,and I is an identity matrix of order 3, show that A3=pI+qA +rA2.

Important Questions on Algebra of Matrices

Learn and Prepare for any exam you want !

If $A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{matrix}]$ ,and $I$ is an identity matrix of order $3$ , show that $A^{3} = p I + q A + r A^{2}$ .