If A-2-3-5-1451-3-4 and B-1351-3-5-1 35- show that A B-B A-O3 - 3 -

Here,AB=2 3 5 1451 3 4 1351 3 5 135 #8658;AB= 2 3+56+9 1510+15 251+4 5 3 12+15 5 20+25 1 3+43+9 125+15 20 #8658;AB=000000000 #8658;AB=03 #215;3 #160; ...

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Byju's Answer

Standard XII

Mathematics

Selecting Consecutive Terms in A.P

If A=2-3-5-14...

Question

If A = $[\begin{matrix} 2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4 \end{matrix}]$ and B = $[\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}]$ , show that AB = BA = O₃_×3.

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Solution

$Here, A B = [\begin{matrix} 2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4 \end{matrix}] [\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}] \Rightarrow A B = [\begin{matrix} - 2 - 3 + 5 & 6 + 9 - 15 & 10 + 15 - 25 \\ 1 + 4 - 5 & - 3 - 12 + 15 & - 5 - 20 + 25 \\ - 1 - 3 + 4 & 3 + 9 - 12 & 5 + 15 - 20 \end{matrix}] \Rightarrow A B = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] \Rightarrow A B = 0_{3 \times 3} . . . (1) B A = [\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}] [\begin{matrix} 2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4 \end{matrix}] \Rightarrow B A = [\begin{matrix} - 2 - 3 + 5 & 3 + 12 - 15 & 5 + 15 - 20 \\ 2 + 3 - 5 & - 3 - 12 + 15 & - 5 - 15 + 20 \\ - 2 - 3 + 5 & 3 + 12 - 15 & 5 + 15 - 20 \end{matrix}] \Rightarrow B A = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] \Rightarrow B A = 0_{3 \times 3} . . . (2) \Rightarrow A B = B A = 0_{3 \times 3} [From eqs . (1) and (2)]$

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Similar questions

Q. If A =

[\begin{matrix} 2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4 \end{matrix}]

and B =

[\begin{matrix} 2 & - 2 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 2 & - 3 \end{matrix}]

, show that AB = A and BA = B.

Q. If

A = [\begin{matrix} 2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4 \end{matrix}]

, show that A² = A.

Q. If

A = [\begin{matrix} 1 & - 2 & 3 - 4 & 2 & 5 \end{matrix}]

and

B = ⎡ ⎢ ⎣ \begin{matrix} 2 & 3 4 & 5 2 & 1 \end{matrix} ⎤ ⎥ ⎦

. Find

A B

and show that

A B \neq B A

Q. Let

A = [\begin{matrix} 1 & - 2 & 3 3 & 2 & - 1 \end{matrix}]

and

B = ⎡ ⎢ ⎣ \begin{matrix} 2 & 3 - 1 & 2 4 & - 5 \end{matrix} ⎤ ⎥ ⎦

. Find AB and BA and show that

A B \neq B A .

Q. If A =

[\begin{matrix} 0 & c & - b \\ - c & 0 & a \\ b & - a & 0 \end{matrix}]

and B =

[\begin{matrix} a^{2} & a b & a c \\ a b & b^{2} & b c \\ a c & b c & c^{2} \end{matrix}]

, show that AB = BA = O₃_×3.

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If A = 2-3-5-1 4 5 1-3-4 and B = -1 3 5 1-3-5-1 3 5, show that AB = BA = O3×3.

Here,AB=2-3-5-1451-3-4-1351-3-5-135⇒AB=-2-3+56+9-1510+15-251+4-5-3-12+15-5-20+25-1-3+43+9-125+15-20⇒AB=000000000⇒AB=03×3 ...1BA=-1351-3-5-1352-3-5-1451-3-4⇒BA=-2-3+53+12-155+15-202+3-5-3-12+15-5-15+20-2-3+53+12-155+15-20⇒BA=000000000 ⇒BA=03×3 ...2⇒AB=BA=03×3 From eqs. (1) and (2)

If A = $[\begin{matrix} 2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4 \end{matrix}]$ and B = $[\begin{matrix} - 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5 \end{matrix}]$ , show that AB = BA = O₃_×3.