Question

What do you evaluate as a matrix?

Answer 1

This type of problem is based on the concept of matrices. A matrix can have a number of rows and columns arranged in a rectangular array. Each element in the matrix is named uniquely and each element performs its own function. We can subtract and add two matrices of the same order but cannot add or subtract a matrix of different order. But that is not the case in multiplication.

Complete step-by-step answer:
According to the question, we are asked to evaluate a matrix.
A matrix is a rectangular array of elements arranged in columns and rows.
A matrix of order $n\times m$ is

{{a}_{11}} & {{a}_{12}} & {{a}_{13}} & ..... & {{a}_{1n}} \\\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} & ..... & {{a}_{2n}} \\\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} & ..... & {{a}_{3n}} \\\ \vdots & {} & {} & {} & {} \\\ {{a}_{m1}} & {{a}_{m2}} & {{a}_{3m}} & ..... & {{a}_{nm}} \\\ \end{matrix} \right]$$. Here, n is the total number of terms in the rows and m is the total number of terms in the column. For example, a $$2\times 2$$ is $$\left[ \begin{matrix} a & b \\\ c & d \\\ \end{matrix} \right]$$. The matrix with same number of rows and columns are called symmetric matrix. Let us consider a matrix which is not symmetric with order $$3\times 2$$. $$\left[ \begin{matrix} a & b & e \\\ c & d & f \\\ \end{matrix} \right]$$ is a 3 by 2 matrix. Let us now consider the addition and subtraction of matrices. A matrix which has the same order can only be added or subtracted. For example, $$\left[ \begin{matrix} a & b & e \\\ c & d & f \\\ \end{matrix} \right]\pm \left[ \begin{matrix} x & y & z \\\ u & v & w \\\ \end{matrix} \right]=\left[ \begin{matrix} a\pm x & b\pm y & e\pm z \\\ c\pm u & d\pm v & f\pm w \\\ \end{matrix} \right]$$. But, we cannot add or subtract a 2 by 2 and 3 by 2 matrix. When it comes to multiplication of two matrices, we can multiply two matrices with different order with certain conditions. The multiplication of two matrices is in the following way. $$\left[ \begin{matrix} a & b \\\ c & d \\\ \end{matrix} \right]\times \left[ \begin{matrix} x & y \\\ u & v \\\ \end{matrix} \right]=\left[ \begin{matrix} ax+bu & ay+bv \\\ cx+du & cy+dv \\\ \end{matrix} \right]$$ Here, the order of the matrices is the same. If the column of the first matrix is equal to the row of the second matrix, then we can multiply the two matrices. For example, we can multiply the matrices of order $$3\times 2$$ and $$2\times 5$$. But, we cannot multiply the matrices of order $$2\times 5$$ and $$3\times 2$$ since 5 is not equal to 3. When it comes to division of two matrices, we cannot divide two matrices. **Note:** We should not simply multiply the corresponding terms of two matrices as we do for addition and subtraction. If all the term of a matrix have a common number, only then, we can take the number out of the matrix that is $$\left[ \begin{matrix} ka & kb \\\ kc & kd \\\ \end{matrix} \right]=k\left[ \begin{matrix} a & b \\\ c & d \\\ \end{matrix} \right]$$.