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Equal matrix
If two matrices \(A\) and \(B\) are equal, they should satisfy the following conditions.
a. Both the matrices should have the same order
b. Each element of matrix \(A\) be equal to the corresponding elements in matrix \(B\). In other words, \(a_{ij}\) \(=\) \(b_{ij}\) for all values of \(i\) and \(j\).
Example:
Let us now look at an example.
\(A = \begin{bmatrix}
5 & 1\\
0 & 3
\end{bmatrix}\) and \(B = \begin{bmatrix}
1^2 + 2^2 & sin^2 \theta + cos^2 \theta\\
1 + \frac{3}{2} - \frac{5}{2} & 2 + sec^2 \theta - tan^2 \theta
\end{bmatrix}\)
5 & 1\\
0 & 3
\end{bmatrix}\) and \(B = \begin{bmatrix}
1^2 + 2^2 & sin^2 \theta + cos^2 \theta\\
1 + \frac{3}{2} - \frac{5}{2} & 2 + sec^2 \theta - tan^2 \theta
\end{bmatrix}\)
Here, \(A\) \(=\) \(B\)
That is, \(\begin{bmatrix}
5 & 1\\
0 & 3
\end{bmatrix}\) \(=\) \(\begin{bmatrix}
1^2 + 2^2 & sin^2 \theta + cos^2 \theta\\
1 + \frac{3}{2} - \frac{5}{2} & 2 + sec^2 \theta - tan^2 \theta
\end{bmatrix}\)
5 & 1\\
0 & 3
\end{bmatrix}\) \(=\) \(\begin{bmatrix}
1^2 + 2^2 & sin^2 \theta + cos^2 \theta\\
1 + \frac{3}{2} - \frac{5}{2} & 2 + sec^2 \theta - tan^2 \theta
\end{bmatrix}\)
Therefore, \(5 = 1^2 + 2^2\), \(1 = sin^2 \theta + cos^2 \theta\), \(0 = 1 + \frac{3}{2} - \frac{5}{2}\) and \(3 = 2 + sec^2 \theta - tan^2 \theta\).
The negative of a matrix
When we replace every element of \(A\) by its additive inverse, we get the negative of matrix \(A\). Every element of \(–A\) is the negative of the corresponding element of \(A\).
Example:
Let us now look at an example.
If, \(A = \begin{bmatrix}
1 & -2 & -3\\
-4 & 5 & 6
\end{bmatrix}\), then \(-A = \begin{bmatrix}
-1 & 2 & 3\\
4 & -5 & -6
\end{bmatrix}\)
1 & -2 & -3\\
-4 & 5 & 6
\end{bmatrix}\), then \(-A = \begin{bmatrix}
-1 & 2 & 3\\
4 & -5 & -6
\end{bmatrix}\)
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