Let and which of the following is true ?

Engineering

Mathematics

Introduction to Determinants

Question

Let $A = [\begin{array}{lll} 1 2 3 \\ 2 0 5 \\ 0 2 1 \end{array}]$ and $B = [\begin{matrix} 0 \\ - 3 \\ 1 \end{matrix}]$ which of the following is true ?

AX = B has a unique solution

AX = B has exactly three solution

AX = B has infinitely many solutions

AX = B is inconsistent

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Given : $A = [\begin{array}{lll} 1 2 3 \\ 2 0 5 \\ 0 2 1 \end{array}]$

Now, $| A | = 1 |\begin{array}{ll} 0 5 \\ 2 1 \end{array}| - 2 |\begin{array}{ll} 2 3 \\ 2 1 \end{array}|$

= 1(– 10) – 2(2 – 6)

= – 10 – 2(– 4)

= – 10 + 8

= – 2

⇒ |A| = – 2 i.e. |A| ≠ 0

Hence, unique solution

∴ option A i.e. AX = B has unique solution

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