Question

I know that, two matrices will equal if their order are same and their corresponding elements are identical..but how to prove it???​

Answer 1

$\boxed{ \sf \: \purple{To \: prove \: : \: AB \not = \: BA }}$

$\sf \: \mapsto \: Let's \: multiply \: A \times B \: matrix \: to \: find \: AB \: matrix.$

$\begin{gathered} \\ \bf \:A = \left[\begin{array}{cc}5& - 1 \\ 6&7\end{array}\right] \\\end{gathered} \: \: \: \: \: \: \: \begin{gathered} \\ \bf \:B = \left[\begin{array}{cc}2& 1 \\ 3&4\end{array}\right] \\\end{gathered}$

$\begin{gathered} \\ \bf \:AB = \left[\begin{array}{cc}(5 \times 2) + ( - 1 \times 3)& (5 \times 1) + ( - 1 \times 4) \\ (6 \times 2) + (7 \times 3) \: \: \: &(6 \times 1) + (7 \times 4) \: \: \: \end{array}\right] \\\end{gathered} \:$

$\begin{gathered} \\ \bf \:AB = \left[\begin{array}{cc}10 + ( - 3)& 5 + (- 4) \\ 12 + 21 \: &6 + 28 \: \end{array}\right] \\\end{gathered}$

$\begin{gathered} \\ \bf \:AB = \left[\begin{array}{cc}7& 1 \\ 33 \: &34 \: \end{array}\right] \\\end{gathered} \sf \: \longrightarrow \boxed { \red{1}}$

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$\sf \: \mapsto \: Let's \: multiply \: B \times A \: matrix \: to \: find \: BA \: matrix.$

$\begin{gathered} \\ \bf \:B = \left[\begin{array}{cc}2& 1 \\ 3&4\end{array}\right] \\\end{gathered} \: \: \: \: \: \: \: \begin{gathered} \\ \bf \:A = \left[\begin{array}{cc}5& - 1 \\ 6&7\end{array}\right] \\\end{gathered}$

$\begin{gathered} \\ \bf \: BA = \left[\begin{array}{cc}(2\times 5) + ( 1 \times 6)& (2 \times - 1) + ( 1 \times 7) \\ \: \: (3 \times 5) + (4 \times 6) \: \: \: & \: \: (3 \times - 1) + (4\times 7) \: \: \: \end{array}\right] \\\end{gathered} \:$

$\begin{gathered} \\ \bf \:BA = \left[\begin{array}{cc}10 + 6& ( - 2) + 7\\ \: \: 15 + 24 \: & - 3 + 28 \: \end{array}\right] \\\end{gathered}$

$\begin{gathered} \\ \bf \:BA = \left[\begin{array}{cc}16& 5\\ \: \: 39 \: & 25 \: \end{array}\right] \\\end{gathered} \longrightarrow \boxed { \red{2}}$

$\sf \: From \: equation \: \boxed{ \red{ 1}} \: and \: \boxed{\red{ 2 }} \: we \: conclude \: AB \: \not = \: BA.$

$\sf \green{\bigstar} \: \underline{ Conditions \: required \:to \: multiply \: matrix. } \: \green{\bigstar}$

The number of columns in the first matrix must be equal to the number of rows in the second matrix

$\sf \green{\bigstar} \: \underline{ More \: to \: know } \: \green{\bigstar}$

The result matrix will have the number of rows of the first and the number of columns of the second matrix.