Algebric properties of Matrices. Definition. (Equality of Matrices). Two [katex]m \times n[/katex] matrices [katex]A=\left[a_{i j}\right][/katex] and [katex]B=\left[b_y\right][/katex] are equal if and only if [katex]a_y=b_y[/katex] for each i and for cach j. When A and B are equal, we write A=B.
Earn Money online without investment Earn Money online from home without investment Definition. (Zero Matrix). A matrix, whose every element is zero, is called a zero matrix. If it has m rows and n columns, we denote it by [katex]0_{\text {mm }}[/katex] or simply by 0 if there is no danger of ambiguity about the number of its rows and columns. Definition. (Addition of Matrices). Two matrices are said to be conformable for addition when they have the same number of rows and the same number of columns. Thus if [katex]A=\left[a_{1 j}\right][/katex] and [katex]B=\left[b_{i j}\right][/katex] are [katex]m \times n[/katex] matrices over the same ficld [katex]F_{\text {, }}[/katex], then they ean be alled and their sum is the matrix
A+B=\left[a_{j j}+b_{i j}\right]of order [katex]m \times n[/katex]. That is, to find the sum A+B of two matrices A and B of the same order we add their corresponding elements. Definition. (Additive Inverse of a Matrix). Given an [katex]m \times n[/katex] matrix [katex]A=\left[a_{i j}\right][/katex], we define [katex]-A=\left[-a_{i j}\right][/katex]. Thus -A is an [katex]m \times n[/katex] matrix and by definition
\begin{aligned}
A+(-A)=\left[a_{i j}+\left(-a_{i j}\right)\right] &=0-\left[\left(-a_{i j}\right)+a_{i j}\right] \
&=(-A)+A .
\end{aligned}Thus -A is the additive inverse of A. Also, [katex]\theta+A=A+\theta=A[/katex], i.c, [katex]\theta[/katex] is the identity with respect to +. Theorem. For any three [katex]m \times n[/katex] matrices A, B and C, we have A+(B+C)=(A+B)+C (Associative Law for Matrix Addition) Proof. Let [katex]A=\left[a_y\right], B=\left[b_j\right], C=\left[c_y\right][/katex]. Then,
\begin{aligned} A+(B+C) &=\left[a_y\right]+\left[b_y+c_v\right] & & \\
&=\left[a_v+\left(b_y+c_v\right)\right], & & \text { (Definition 3.7) } \\
&=\left[\left(a_y+b_y\right)+c_v\right], & &\text { (Associative Law in } F) \\ &=\left[a_y+b_v\right]+\left[c_v\right], & & \text { (Definition 3.7) } \end{aligned}Definition. (Subtraction of Matrices). If A and B are two matrices of the gnte order, we define A-B as A-B=A+(-B). For example, let [katex]…[/katex]
\begin{aligned}
A &=\left[\begin{array}{rrr}
2 & 1 & -3 \\
-4 & 0 & 1
\end{array}\right], B=\left[\begin{array}{rrr}
1 & -1 & 2 \\
0 & 1 & 0
\end{array}\right] \
\end{aligned}Then A-B=A+(-B)
\begin{aligned}
&=\left[\begin{array}{rrr}
2 & 1 & -3 \\
-4 & 0 & 1
\end{array}\right]+\left[\begin{array}{rrr}
-1 & 1 & -2 \\
0 & -1 & 0
\end{array}\right] \\
&=\left[\begin{array}{rrr}
2-1 & 1+1 & -3-2 \\
-4+0 & 0-1 & 1+0
\end{array}\right]=\left[\begin{array}{rrr}
1 & 2 & -5 \\
-4 & -1 & 1
\end{array}\right]
\end{aligned}