PDF chapter test

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 $$B = \begin{bmatrix} 1^2 + 2^2 & sin^2 \theta + cos^2 \theta\\ 1 + \frac{3}{2} - \frac{5}{2} & 2 + sec^2 \theta - tan^2 \theta \end{bmatrix}$$

Here, $$A$$ $$=$$ $$B$$

That is, $$\begin{bmatrix} 5 & 1\\ 0 & 3 \end{bmatrix}$$ $$=$$ $$\begin{bmatrix} 1^2 + 2^2 & sin^2 \theta + cos^2 \theta\\ 1 + \frac{3}{2} - \frac{5}{2} & 2 + sec^2 \theta - tan^2 \theta \end{bmatrix}$$

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 $$A$$.
The negative of matrix $$A_{m \times n}$$ is denoted by $$-A_{m \times n}$$.
Example:
Let us now look at an example.

If, $$A = \begin{bmatrix} 1 & -2 & -3\\ -4 & 5 & 6 \end{bmatrix}$$, then $$-A = \begin{bmatrix} -1 & 2 & 3\\ 4 & -5 & -6 \end{bmatrix}$$