Let A - begin bmatrix 28311182 e n d b m a t r i x and - B - A - begin bmatrix 4 -6 -2 - 4

Here A = [ 2 amp; 3; 1 amp; 2 ]and B= [ 4 amp; 6; 2 amp; 4 ] ⇒ AB=[ 2 amp; 3; 1 amp; 2 ][ 4 amp; 6; 2 amp; ...

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Standard XII

Mathematics

Matrix Definition and Representation

Let A = begin...

Question

Let $A = [\begin{matrix} 2 & 3 1 & 2 \end{matrix}] and B = A = [\begin{matrix} 4 & - 6 - 2 & 4 \end{matrix}] . Then compute AB. Hence, solve the following system of equations : 2x + y = 4, 3x + 2y = 1 .$

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Solution

Here $A = [\begin{matrix} 2 & 3 1 & 2 \end{matrix}] and B = [\begin{matrix} 4 & - 6 - 2 & 4 \end{matrix}] \Rightarrow A B = [\begin{matrix} 2 & 3 1 & 2 \end{matrix}] [\begin{matrix} 4 & - 6 - 2 & 4 \end{matrix}] = [\begin{matrix} 2 & 0 0 & 2 \end{matrix}] = 2 I$
$\Rightarrow A (\frac{1}{2} B) = I ∴ A^{- 1} = \frac{1}{2} B . . . . . (i) The given system of equations 2 x + y = 4, 3 x + 2 y = 1 can be written as P X = C w h e r e, P = [\begin{matrix} 2 & 1 3 & 2 \end{matrix}], X = [\begin{matrix} x y \end{matrix}] and C = [\begin{matrix} 41 \end{matrix}] \Rightarrow P^{- 1} P X = P^{- 1} C \Rightarrow I X = P^{- 1} C ∴ X = P^{- 1} C [Note that P = A^{T} ∴ P^{- 1} = (A^{- 1})^{T}] Therefore X = \frac{1}{2} [\begin{matrix} 4 & - 2 - 6 & 4 \end{matrix}] [\begin{matrix} 41 \end{matrix}] \Rightarrow X = \frac{1}{2} [\begin{matrix} 14 - 20 \end{matrix}] \Rightarrow [\begin{matrix} x y \end{matrix}] = [\begin{matrix} 7 - 10 \end{matrix}]$
By equality of matrices, we get : x = 7, y = -10.

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Let A=[2312]and B=A=[4−6−24].Then compute AB. Hence, solve the following system of equations : 2x + y = 4, 3x + 2y = 1.

Let $A = [\begin{matrix} 2 & 3 1 & 2 \end{matrix}] and B = A = [\begin{matrix} 4 & - 6 - 2 & 4 \end{matrix}] . Then compute AB. Hence, solve the following system of equations : 2x + y = 4, 3x + 2y = 1 .$