alexoooo137
Answered

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Q). If
| 1 , 1, 3 |
A = | 5, 2, 6 |, then find A³
| -2, -1, -3 |​

Q If 1 1 3 A 5 2 6 Then Find A 2 1 3 class=

Sagot :

felixcharles333

Answer:

A³ = [ 1, 1, 27, 125, 8, 216, -8, -1, -27].

Step-by-step explanation:

All the numbers in the bracket should be cubed.

I hope you understood ♥️♥️♥️

SalmonWorldTopper

Step-by-step explanation:

[tex]\large\underline{\sf{Solution-}}[/tex]

Given matrix is

[tex]\rm :\longmapsto\: \begin{gathered}\sf A=\left[\begin{array}{ccc}1&1&3\\5&2&6\\ - 2&-1& - 3\end{array}\right]\end{gathered}[/tex]

Consider,

[tex]\red{\rm :\longmapsto\: {A}^{2}}[/tex]

[tex]\rm \:  =  \: A \times A[/tex]

[tex] \\ \rm \:  =  \: \begin{gathered}\sf \left[\begin{array}{ccc}1&1&3\\5&2&6\\ - 2&-1& - 3\end{array}\right]\end{gathered}\begin{gathered}\sf \left[\begin{array}{ccc}1&1&3\\5&2&6\\ - 2&-1& - 3\end{array}\right]\end{gathered} \\ [/tex]

[tex]\rm \:  =  \: \begin{gathered}\sf \left[\begin{array}{ccc}1 + 5 - 6&1 + 2 - 3&3 + 6 - 9\\5 + 10 - 12&5 + 4 - 6&15 + 12 - 18\\ - 2 - 5 + 6&-2 - 2 + 3& - 6 - 6 + 9\end{array}\right]\end{gathered}[/tex]

[tex]\rm \:  =  \: \begin{gathered}\sf \left[\begin{array}{ccc}0&0&0\\3&3&9\\ - 1&-1& - 3\end{array}\right]\end{gathered}[/tex]

[tex] \\ \bf\implies \: {A}^{2} = \begin{gathered}\sf \left[\begin{array}{ccc}0&0&0\\3&3&9\\ - 1&-1& - 3\end{array}\right]\end{gathered} \\ [/tex]

Now, Consider

[tex]\red{\rm :\longmapsto\: {A}^{3}}[/tex]

[tex]\rm \:  =  \: {A}^{2} \times A[/tex]

[tex] \\ \rm \:  =  \: \begin{gathered}\sf \left[\begin{array}{ccc}0&0&0\\3&3&9\\ - 1&-1& - 3\end{array}\right]\end{gathered} \times \begin{gathered}\sf \left[\begin{array}{ccc}1&1&3\\5&2&6\\ - 2&-1& - 3\end{array}\right]\end{gathered}[/tex]

[tex] \\ \rm \:  =  \: \begin{gathered}\sf \left[\begin{array}{ccc}0&0&0\\3 + 15 - 18&3 + 6 - 9&9 + 18 - 27\\ - 1 - 5 + 6&-1 - 2 + 3& - 3 - 6 + 9\end{array}\right]\end{gathered} \\ [/tex]

[tex] \\ \rm \:  =  \: \begin{gathered}\sf \left[\begin{array}{ccc}0&0&0\\0&0&0\\ 0&0&0\end{array}\right]\end{gathered}[/tex]

[tex] \\ \bf\implies \: {A}^{3}   =  \: \begin{gathered}\sf \left[\begin{array}{ccc}0&0&0\\0&0&0\\ 0&0&0\end{array}\right]\end{gathered}[/tex]

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Learn More :-

Matrix multiplication is defined when number of columns of pre multiplier is equal to the number of rows of post multiplier.

Matrix multiplication may or may not be Commutative.

Matrix multiplication is Associative. i.e (AB)C = A(BC)

Matrix multiplication is Distributive. i.e. A ( B + C ) = AB + AC

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