Question

If A = $\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{bmatrix}$ then A³ – rA² – qA =

• A. p I
• B. q I
• C. r I
• D. None of these

Answer 1

Answer: (A)

p I

Answer 2

A² = AA = $\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{bmatrix}$ × $\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{bmatrix}$= $\begin{bmatrix} 0 & 0 & 1 \\ p & q & r \\ pr & p + qr & q + r^{2} \end{bmatrix}$Again A³ = A²A

=$\begin{bmatrix} 0 & 0 & 1 \\ p & q & r \\ pr & p + qr & q + r^{2} \end{bmatrix}$ × $\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{bmatrix}$

= $\begin{bmatrix} p & q & r \\ pr & p + qr & q + r^{2} \\ pq + r^{2}p & pr + q^{2} + qr^{2} & p + 2qr + r^{3} \end{bmatrix}$

= $\begin{bmatrix} p & 0 & 0 \\ 0 & p & 0 \\ 0 & 0 & p \end{bmatrix}$ + $\begin{bmatrix} 0 & q & 0 \\ 0 & 0 & q \\ pq & q^{2} & qr \end{bmatrix}$ + $\begin{bmatrix} 0 & 0 & r \\ pr & qr & r^{2} \\ pr^{2} & pr + qr^{2} & qr + r^{3} \end{bmatrix}$

= p $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ + q $\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{bmatrix}$

+ r $\begin{bmatrix} 0 & 0 & 1 \\ p & q & r \\ pr & p + qr & q + r^{2} \end{bmatrix}$

= pI + qA + rA²

\ a³ – rA² – qA = pI.