Question

Let A =$\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}$. If U₁, U₂, U₃ are column matrices

satisfying AU₁ = $\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, AU₂ = $\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$, AU₃ = $\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}$ and U is

3 × 3 matrix whose columns are U₁, U₂ and U₃. Then |U| =

• A. 3
• B. 3/2
• C. –3
• D. 2

Answer 1

Answer: (A)

3

Answer 2

$\begin{bmatrix} a_{1} \\ b_{1} \\ c_{1} \end{bmatrix}$, U₂ = $\begin{bmatrix} a_{2} \\ b_{2} \\ c_{2} \end{bmatrix}$, U₃ = $\begin{bmatrix} a_{3} \\ b_{3} \\ c_{3} \end{bmatrix}$

A. U₁ = $\left[ \begin{array} { l l l } 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{array} \right]$ $\begin{bmatrix} a_{1} \\ b_{1} \\ c_{1} \end{bmatrix}$ = $\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$

=$\begin{bmatrix} a_{1} \\ 2a_{1} + b_{1} \\ 3a_{1} + 2b_{1} + c_{1} \end{bmatrix}$=$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$

̃ a₁ = 1, b₁ = –2, c₁ = 1

̃ U₁ = $\begin{bmatrix} 1 \\ –2 \\ 1 \end{bmatrix}$

Similarly U₂ = $\begin{bmatrix} 2 \\ –1 \\ –4 \end{bmatrix}$& U₃ = $\begin{bmatrix} 2 \\ –1 \\ –3 \end{bmatrix}$

U = $\begin{bmatrix} 1 & 2 & 2 \\ –2 & –1 & –1 \\ 1 & –4 & –3 \end{bmatrix}$ ̃ |U| = 3