# Section5.3Laws of Matrix Algebra¶ permalink

The following is a summary of the basic laws of matrix operations. Assume that the indicated operations are defined; that is, that the orders of the matrices $A\text{,}$ $B$ and $C$ are such that the operations make sense.

##### Example5.3.2More Precise Statement of one Law

If we wished to write out each of the above laws more completely, we would specify the orders of the matrices. For example, Law 10 should read:

Let $A\text{,}$ $B\text{,}$ and $C$ be $m\times n$, $n\times p$, and $n\times p$ matrices, respectively, then $A(B + C) = A B + A C$

Remarks:

• Notice the absence of the “law” $A B = B A$. Why?

• Is it really necessary to have both a right (No. 11) and a left (No. 10) distributive law? Why?

##### 1

Rewrite the above laws specifying as in Example 5.3.2 the orders of the matrices.

##### 2

Verify each of the Laws of Matrix Algebra using examples.

##### 3

Let $A = \left( \begin{array}{cc} 1 & 2 \\ 0 & -1 \\ \end{array} \right)$, $B= \left( \begin{array}{ccc} 3 & 7 & 6 \\ 2 & -1 & 5 \\ \end{array} \right)$, and $C= \left( \begin{array}{ccc} 0 & -2 & 4 \\ 7 & 1 & 1 \\ \end{array} \right)$. Compute the following as efficiently as possible by using any of the Laws of Matrix Algebra:

1. $A B + A C$

2. $A^{-1}$

3. $A(B + C)$

4. $\left(A^2\right)^{-1}$

5. $(C + B)^{-1}A^{-1}$

##### 4

Let $A =\left( \begin{array}{cc} 7 & 4 \\ 2 & 1 \\ \end{array} \right)$ and $B =\left( \begin{array}{cc} 3 & 5 \\ 2 & 4 \\ \end{array} \right)$. Compute the following as efficiently as possible by using any of the Laws of Matrix Algebra:

1. $A B$

2. $A + B$

3. $A^2 + A B + B A + B ^2$

4. $B^{-1}A^{-1}$

5. $A^2 + A B$

##### 5

Let $A$ and $B$ be $n\times n$ matrices of real numbers. Is $A^2-B^2= (A-B)(A+B)$? Explain.