If A -1- bmatrix 38-1 - 111-15 - 68-5 15 -2 - 2end bmatrix and B - begin bmatrix 182 -2-1 - 3 - 0 10 -2 - 1end bmatrix - find AB -1-

We know that, (AB)^ 1=B^ 1A^ 1 and A^ 1 is known, therefore we proceed to find B^ 1 Here, |B|= 1 amp;2 amp; 2 1 amp;3 amp;0 0 amp; 2 ...

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Standard XII

Mathematics

Difference of Two Sets

If A ∧-1=. bm...

Question

If $A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦$ and $B = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & - 2 - 1 & 3 & 0 0 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦$ , find $(A B)^{- 1}$ .

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Solution

We know that, $(A B)^{- 1} = B^{- 1} A^{- 1} a n d A^{- 1}$ is known, therefore we proceed to find $B^{- 1}$
Here, $| B | = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & - 2 - 1 & 3 & 0 0 & - 2 & 1 \end{matrix} ∣ ∣ ∣ ∣ = 1 (3 - 0) - 2 (- 1 - 0) - 2 (2 - 0) = 3 + 2 - 4 = 1 \neq 0$
$∴ B^{- 1}$ exists. Cofactors of B are
$\begin{matrix} B_{11} = (3 - 0) = 3, & B_{12} = - (- 1 - 0) = 1, & B_{13} = (2 - 0) = 2 B_{21} = - (2 - 4 = 2), & B_{22} = 1 - 0 = 1, & B_{23} = - (- 2 - 0) = 2 B_{31} = (0 + 6) = 6 & B_{32} = - (0 - 2) = 2, & B_{33} = 3 + 2 = 5 \end{matrix}$
$∴ a d j (B) = {⎡ ⎢ ⎣ \begin{matrix} 3 & 1 & 2 2 & 1 & 2 6 & 2 & 5 \end{matrix} ⎤ ⎥ ⎦}^{T} = ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 6 1 & 1 & 2 2 & 2 & 5 \end{matrix} ⎤ ⎥ ⎦$
$∴ B^{- 1} = \frac{1}{| B |} (a d j (B)) = \frac{1}{1} ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 6 1 & 1 & 2 2 & 2 & 5 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 6 1 & 1 & 2 2 & 2 & 5 \end{matrix} ⎤ ⎥ ⎦$
Now, $(A B)^{- 1} = B^{- 1} A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 3 & 2 & 6 1 & 1 & 2 2 & 2 & 5 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦$
$= ⎡ ⎢ ⎣ \begin{matrix} 9 - 30 + 30 & - 3 + 12 - 12 & 3 - 10 + 12 3 - 15 + 10 & - 1 + 6 - 4 & 1 - 5 + 4 6 - 30 + 25 & - 2 + 12 - 10 & 2 - 10 + 10 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 9 & - 3 & 5 - 2 & 1 & 0 1 & 0 & 2 \end{matrix} ⎤ ⎥ ⎦$

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Q. If

A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦

and

B = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & - 2 - 1 & 3 & 0 0 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦

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{(A B)}^{- 1}

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 1 0 & 1 & - 1 3 & - 1 & 1 \end{matrix} ⎤ ⎥ ⎦

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Q. Find

(A B)^{- 1}

, if

A = [\begin{matrix} 1 & 2 & 3 1 & - 2 & - 3 \end{matrix}], B = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 1 & 2 1 & - 2 \end{matrix} ⎤ ⎥ ⎦

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If A−1=⎡⎢⎣3−11−156−55−22⎤⎥⎦ and B=⎡⎢⎣12−2−1300−21⎤⎥⎦, find (AB)−1.

If $A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 3 & - 1 & 1 - 15 & 6 & - 5 5 & - 2 & 2 \end{matrix} ⎤ ⎥ ⎦$ and $B = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & - 2 - 1 & 3 & 0 0 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦$ , find $(A B)^{- 1}$ .