Important: We can only multiply matrices if the number of columns in the first matrix is the exact same as the variety of rows in the 2nd matrix.

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Example 1

a) multiplying a 2 × 3 matrix by a 3 × 4 matrix is possible and it provides a 2 × 4 matrix as the answer.

b) multiplying a 7 × 1 procession by a 1 × 2 matrix is okay; it provides a 7 × 2 matrix

c) A 4 × 3 procession times a 2 × 3 procession is not possible.

How to main point 2 Matrices

We use letters an initial to view what is going on. We"ll view a numbers example after.

As an example, let"s take it a basic 2 × 3 procession multiplied by a 3 × 2 matrix.

`<(a,b,c),(d,e,f)><(u,v),(w,x),(y,z)>`

The answer will certainly be a 2 × 2 matrix.

We main point and include the elements as follows. We work-related across the first row that the first matrix, multiplying down the 1st column of the second matrix, element by element. Us add the resulting products. Our answer go in position a11 (top left) of the price matrix.

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We do a similar procedure for the 1st row that the very first matrix and also the 2nd tower of the 2nd matrix. The result is put in place a12.

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Now for the 2nd row of the first matrix and also the 1st tower of the 2nd matrix. The result is placed in place a21.

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Finally, we carry out the 2nd row that the very first matrix and the second column of the second matrix. The result is put in place a22.

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So the an outcome of multiplying our 2 matrices is as follows:

`<(a,b,c),(d,e,f)><(u,v),(w,x),(y,z)>` `=<(au+bw+cy,av+bx+cz),(du+ew+fy,dv+ex+fz)>`

Now let"s see a number example.


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Matrices and also Systems the Simultaneous linear Equations

We now see exactly how to write a device of linear equations making use of matrix multiplication.

Example 4

The system of equations

−3x + y = 1

6x − 3y = −4

can be written as:

`((-3,1),(6,-3))((x),(y))=((1),(-4))`

Matrices are appropriate for computer-driven services of problems because computers easily type arrays. We have the right to leave the end the algebraic symbols. A computer only requires the an initial and critical matrices to fix the system, as we will view in Matrices and Linear Equations.

Note 1 - Notation

Care v writing procession multiplication.

The complying with expressions have various meanings:

AB is matrix multiplication

A×B is cross product, which return a vector

A*B offered in computer notation, however not top top paper

A•B dot product, which return a scalar.

Note 2 - Commutativity of matrix Multiplication

Does `AB = BA`?

Let"s see if that is true using an example.

Example 5

If

`A=((0,-1,2),(4,11,2))`

and

`B=((3,-1),(1,2),(6,1))`

find ab and BA.

Answer


We performed abdominal muscle above, and the answer was:


`AB = ((0,-1,2),(4,11,2)) ((3,-1),(1,2),(6,1))`

` = ( (11,0),(35,20) )`


Now BA is (3 × 2)(2 × 3) i m sorry will give 3 × 3:


`BA= ((3,-1),(1,2),(6,1))((0,-1,2),(4,11,2))`

` = ((0-4,-3-11,6-2),(0+8,-1+22,2+4),(0+4,-6+11,12+2))`

` = ((-4,-14,4),(8,21,6),(4,5,14)) `


So in this case, AB does no equal BA.

In fact, for most matrices, girlfriend cannot reverse the stimulate of multiplication and also get the very same result.


In general, when multiplying matrices, the commutative regulation doesn"t hold, i.e. ABBA. There are two usual exceptions to this:

The inverse of a matrix: A-1A = AA-1 = I.

In the next section we learn exactly how to uncover the station of a matrix.

Example 6 - multiplying by the identity Matrix

Given that

`A=((-3,1,6),(3,-1,0),(4,2,5))`

find AI.

Answer


`AI = ((-3,1,6),(3,-1,0),(4,2,5)) ((1,0,0),(0,1,0),(0,0,1))`

`=((-3+0+0,0+1+0,0+0+6),(3+0+0,0+ -1+0,0+0+0),(4+0+0,0+2+0,0+0+5))`

` =((-3,1,6),(3,-1,0),(4,2,5))`

` =A`


We see that multiply by the identity matrix go not change the value of the original matrix.

That is,

AI = A


Exercises

1. If possible, discover BA and AB.

`A=((-2,1,7),(3,-1,0),(0,2,-1))`

`B=(4 -1 5)`

Answer


`BA=(4 -1 5)((-2,1,7),(3,-1,0),(0,2,-1))`

`=( -8+(-3)+0 4+1+10 28+0+(-5))`

`=(-11 15 23)`


AB is no possible. (3 × 3) × (1 × 3).


2. Recognize if B = A-1, given:

`A=((3,-4),(5,-7))`

`B=((7,4),(5,3))`

Answer


If B = A-1, climate `AB = I`.


`AB=((3,-4),(5,-7))((7,4),(5,3))`

`=((21-20,12-12),(35-35,20-21))`

`=((1,0),(0,-1))`

` !=I`


So B is not the train station of A.


3. In researching the movement of electrons, among the Pauli turn matrices is

`s=((0,-j),(j,0))`

where

`j=sqrt(-1)`

Show the s2 = I.

.

Answer


`s^2=( (0,-j),(j,0))((0,-j),(j,0))`

`=(( 0-j^2,0+0), (0+0,-j^2+0))`

`= ((1,0),(0,1))`

`=I`


4. Advice the following matrix multiplication i beg your pardon is used in directing the motion of a robotic mechanism.


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`( (cos 60° ,-sin 60° ,0),(sin 60°, cos 60°,0),(0,0,1))((2),(4),(0))`

Answer


`( (cos 60° ,-sin 60° ,0),(sin 60°, cos 60°,0),(0,0,1))((2),(4),(0))`

`=((2(0.5)-4(0.866)+0),(2(0.866)+4(0.5)+0),(0+0+0))`

`=((-2.464),(3.732),(0))`

The translate of this is the the robot arm moves from place (2, 4, 0) to position (-2.46, 3.73, 0). That is, it move in the x-y plane, yet its height remains in ~ z = 0.

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The 3 × 3 procession containing sin and cos values tells the how countless degrees to move.