Let and U3 be column matrices satisfying . If U is 3 × 3 matrix, whose columns are U1, U2, U3. then |U| is

Engineering

Mathematics

Question

Let $A = [\begin{array}{lll} 1 0 0 \\ 2 1 0 \\ 3 2 1 \end{array}], U_{1}, U_{2}$ and U₃ be column matrices satisfying ${AU}_{1} = [\begin{array}{c} 1 \\ 0 \\ 0 \end{array}], {AU}_{2} = [\begin{array}{c} 2 \\ 3 \\ 6 \end{array}], {AU}_{3} = [\begin{array}{c} 2 \\ 3 \\ 1 \end{array}]$ . If U is 3 × 3 matrix, whose columns are U₁, U₂, U₃. then |U| is

– 11

– 3

$\frac{3}{2}$

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$A = [\begin{array}{lll} 1 0 0 \\ 2 1 0 \\ 3 2 1 \end{array}]$

A^–1 = ?

|A| = 1(1 – 0) – 0 + 0

= 1

$A^{⊤} = [\begin{array}{lll} 1 2 3 \\ 0 1 2 \\ 0 0 1 \end{array}]$

$Adj (A^{⊤}) = [\begin{array}{ccc} 1 & 0 & 0 \\ - 2 & 1 & 0 \\ 1 & - 2 & 1 \end{array}]$

$∴ A^{- 1} = \frac{1}{| A |} Adj [A^{⊤}]$

$= [\begin{array}{ccc} 1 & 0 & 0 \\ - 2 & 1 & 0 \\ 1 & - 2 & 1 \end{array}]$

$∴ U_{1} = [\begin{array}{l} 1 \\ 0 \\ 0 \end{array}] [\begin{array}{ccc} 1 & 0 & 0 \\ - 2 & 1 & 0 \\ 1 & - 2 & 1 \end{array}]$

$= [\begin{matrix} 1 \\ - 2 \\ 1 \end{matrix}]$

$U_{2} = [\begin{array}{l} 2 \\ 3 \\ 6 \end{array}] [\begin{array}{ccc} 1 & 0 & 0 \\ - 2 & 1 & 0 \\ 1 & - 2 & 1 \end{array}]$

$= [\begin{array}{l} 2 \\ - 4 + 3 + 0 \\ 02 - 6 + 6 \end{array}]$

$= [\begin{matrix} 2 \\ - 1 \\ 2 \end{matrix}]$

$U_{3} = [\begin{array}{l} 2 \\ 3 \\ 1 \end{array}] [\begin{array}{ccc} 1 & 0 & 0 \\ - 2 & 1 & 0 \\ 1 & - 2 & 1 \end{array}]$

$= [\begin{matrix} 2 \\ - 4 + 3 \\ 2 - 6 + 1 \end{matrix}] = [\begin{matrix} 2 \\ - 1 \\ - 3 \end{matrix}]$

$| U | = |\begin{array}{ccc} 1 & 2 & 2 \\ - 2 & - 1 & - 1 \\ 1 & 2 & - 3 \end{array}|$

= 1(3 + 2) – 2(6 + 1) – 2(– 4 + 1)

= 5 – 14 + 6

= – 3