Explain why the formula is not valid for matrices. Illustrate your argument with examples. (A+B)(A+B)=A^2+2AB+B^2

Explain why the formula is not valid for matrices. Illustrate your argument with examples. (A+B)(A+B)=A^2+2AB+B^2

asked 2020-11-26
Explain why the formula is not valid for matrices. Illustrate your argument with examples.

Answers (1)

Step 1
Consider the provided question,
According to you we have to solve only 38 question.
Show that why the formula \((A+B)(A+B)=A^2+2AB+B^2\) is not valid for matrices. In the case of matrices,
\(= A^2+AB+BA+B^2\)
Since, we know that hhe relation AB=BA is not always correct.
Step 2
Now, consider an example.
Let \(A=\begin{bmatrix}1 & 2 \\ 0 & 3 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -1 \\ 1 & 2 \end{bmatrix}\)
\(AB=\begin{bmatrix}1 & 2 \\ 0 & 3 \end{bmatrix}\begin{bmatrix}0 & -1 \\ 1 & 2 \end{bmatrix}\)
\(=\begin{bmatrix}1\cdot0+2\cdot1 & 1(-1)+2\cdot2 \\ 0\cdot0+3\cdot1 & 0(-1)+3\cdot2 \end{bmatrix}\)
\(=\begin{bmatrix}2& 3 \\3 & 6 \end{bmatrix}\)
Step 3
\(A=\begin{bmatrix}1 & 2 \\ 0 & 3 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -1 \\ 1 & 2 \end{bmatrix}\)
\(BA=\begin{bmatrix}0 & -1 \\ 1 & 2 \end{bmatrix}\begin{bmatrix}1 & 2 \\ 0 & 3 \end{bmatrix}\)
\(=\begin{bmatrix}0\cdot1+(-1)0 & 0\cdot2+(-1)3 \\1\cdot1+2\cdot0 & 1\cdot2+2\cdot3 \end{bmatrix}\)
\(=\begin{bmatrix}0 & -3 \\1 & 8 \end{bmatrix}\)
Here, \(AB \neq BA\)
and as a result \(AB+BA \neq 2AB\)
Thus, the formula \((A+B)(A+B)=A^2+2AB+B^2\) is not correct for all matrices.

Relevant Questions

asked 2020-12-02
Explain why each of the following algebraic rules will not work in general when the real numbers a and b are replaced by \(n \times n\) matrices A and B. \(a) (a+b)^2=a^2+2ab+b^2\)
\(b) (a+b)(a-b)=a^2-b^2\)
asked 2021-02-08
Determine whether each statement is sometimes, always , or never true for matrices A and B and explain your reasoning. Stuck on one of them? Try a few examples.
a) If A+B exists , then A-B exists
b) If A and B have the same number of elements , then A+B exists.
asked 2020-11-30
To illustrate the multiplication of matrices, and also the fact that matrix multiplication is not necessarily commutative, consider the matrices
\(A=\begin{bmatrix}1 & -2&1 \\0 & 2&-1\\2&1&1 \end{bmatrix}\)
\(B=\begin{bmatrix}2 & 1&-1 \\1 & -1&0\\2&-1&1 \end{bmatrix}\)
asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
asked 2021-02-25
We will now add support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be restricted to register indirect (i.e., all addresses are simply a value held in a register; no offset or displacement may be added to the register value). For example, the register-memory instruction add x4, x5, (x1) means add the contents of register x5 to the contents of the memory location with address equal to the value in register x1 and put the sum in register x4. Register-register ALU operations are unchanged. The following items apply to the integer RISC pipeline:
a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.
b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.
c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.
d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.
Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.
asked 2021-01-04
Matrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if
\(A=\begin{bmatrix}2&1&1 \\-1&-1&4 \end{bmatrix} , B=\begin{bmatrix}0 & 2 \\-4 & 1\\2&-3 \end{bmatrix} , C=\begin{bmatrix}6 & -1 \\3 & 0\\-2&5 \end{bmatrix} , D=\begin{bmatrix}2 & -3&4 \\-3& 1&-2 \end{bmatrix}\)
If the operation is not possible , write NOT POSSIBLE and be able to explain why
asked 2020-11-30
Explain why S is not a basis for \(M_{2,2}\)
\(S=\left\{\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix},\begin{bmatrix}0 & 1\\1 & 0 \end{bmatrix},\begin{bmatrix}1 & 1 \\0 & 0 \end{bmatrix}\right\}\)
asked 2020-11-10
If A and B are nn matrices, then \((A-B)^2 =A^2-2AB+B^2\)
asked 2021-02-19
Determine whether each of the following statements is true or false, and explain why.If A and B are square matrices of the same size, then AB = BA
asked 2021-02-08
Let \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) be the set of 2 x 2 matrices with the entries in \(\mathbb{Z}/\mathbb{6Z}\)
a) Can you find a matrix \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) whose determinant is non-zero and yet is not invertible?
b) Does the set of invertible matrices in \(M_{2 \times 2} (\mathbb{Z}/\mathbb{6Z})\) form a group?