Suppose you're trying to multiply matrix A, which has m rows and n columns, by matrix B, which has n rows and p columns. AB m xn n x p

Note that the number of columns in the left-hand matrix must equal the number if rows in the right-hand matrix; otherwise, it is not possible to multiply the matrices.
Here is the procedure to find the matrix product:
1. Start with the first row of A and the first column of B. (Note that both of these will contains n elements.)
2. Multiply the corresponding elements of this row ad this column.
3. Add all the products together. The result will be the element in the first row and first column of the product matrix C (AB).
4. Repeat this process to find each element of AB, using the ith row A and the jth column of B to get the element in the ith row and the jth column of AB. (the resulting matrix will have m rows and p columns.)

For example:
Multiplying a 3-by-3 matrix A times a 3-by-1 matrix B: A X B
[3 7 11] X [20]
[9 4 8] X [40]
[5 6 10] X [30]

We started by taking the first row of A and the first (in this case, only) column of B.
3 x 20 + 7 x 40 + 11 x 30 = 670

The result of this calculation was the element in row 1, column 1 of the product matrix C (AB). To find the element in row 2, column 1 of the product matrix, we needed to take the second row of A and the first column of B:
9 x 20 + 4 x 40 + 8 x 30 = 580

Then we needed to that the third row of A and the first column of B:
5 x 20 + 6 x 40 + 10 x 30 = 640

Therefore we could write the complete product matrix:
C(AB)=
[3 7 11] X [20] [670]
[9 4 8] X [40] = [580]
[5 6 10] X [30] [640]

Donald Taylor and Sadia Akhtar
Guidelines for Multiplying Matrices

When multiplying matrices you must multiply the same number of rows in the first matrix by the same number of columns in the second. The insides of the equation much match. For example a 2x1 matrix by a 1x1 matrix will work because you have 1 row in the first matrix and 1 column in the second. If you try to multiply a 1x1 matrix by a 2x1 matrix it won't work because you have 1 row in the first matrix and 2 columns in the second. They must match. This is also why matrices aren't commutative because if you flip them they may not work. You also can't multiply a 2 by 1 matrix with a 1 by 2 matrix, but you can multiply a 1 by 2 matrix with a 2 by 1 matrix.

Robert Yemola
Warm Up From 4-1-08

1. Is matrix multiplication commutative? Prove using examples.

No they are not because in the example the below the products are not equal.

1 2 * 5 6 = 19 22
3 4 * 7 8 = 43 50

5 6 * 1 2 = 23 34
7 8 * 3 4 = 31 46

Joshua Hendarto

To determine the answer of the multiplied matrix, its always the outside column times the number of rows

Ex:
Matrix A = 3 by 4
Matrix B = 4 by 6

Answer = 3 by 6

Melanie Thomas*
*When multiplying matrices, the insides must match up. For example: you can multiply a 1x2 matrix by a 2x1 matrix and the result will be a matrix with the outside values. Example: The result would be a 1x1 matrix.

A matrix is a rectangular array of numbers or algebraic expressions enclosed in square brackets. They are denoted by a capital letter. There are rows and columns. If matrix m has 2 rows 3 columns, the dimensions of m are 2x3. M2,1 is the entry in the second row, first column. m r,c is the rth row and the cth column.

Michael Miller
Matrices are used to describe linear equations of coefficients of linear transformations and to record data.

Milana Lewis-Zakuto
A matrix is a an algebraic expression enclosed in square brackets (Some math texts use parentheses instead.) Usually a matrix has rows and columns and in mathematics a row is horizontal and a column is vertical.s

Dakota Townsend
Matrix: An arrangement of data in rows and columns and enclosed brackets[ ].
Matrix Equality: Two matrices are equal when their dimensions are the same and their corresponding entries are equal.

Adding and Subtracting Matrices:
If you have two matrices A and B and if they have the same dimensions than if you add them together the contents inside are the sum of the numbers. Also the dimensions will remain the same.
Example-

1.

2.

Matrices that Represent Systems of Equations:
Sometimes you may get a problem that has 3 different equations in it, and instead of trying to solve them all the regular way you can use a matrix to make it easier.
Example-

Now you can take these equations and make it into a matrix by copying the numbers into the brackets.

A- coefficients
X- variables
B- constants

Multiplying Matrices:
When you multiply matrices you end up with one that is called the product matrix. For the matrix to work the number of columns in the 1rst matrix must be equal to the number of rows in the 2nd matrix.
Now when you do this you have to multiply first than add.
For example-

3.

4.

Inverse of a Matrix:
When finding the inverse of a matrix you must know the identity of it. Usually the identity of a matrix depends on the dimensions.If you have a 3x3 matrix the identity matrix will also be a 3x3 but it will look like this:

Also when making the identity matrix all the 1's go diagonal while the 0's go everywhere else. Basically your goal is to get the identity matrix on the left side.

Erik Duarte:

Classwork for March 31, 2008

Matrix Multiplication

LS-61 c

Ipiphani Battle March 31, 2008

General Defintion of Matrix Multiplication

Suppose you're trying to multiply matrix A, which has

mrows andncolumns, by matrix B, which hasnrows andpcolumns.ABm xn n x pNote that the number of columns in the left-hand matrix must equal the number if rows in the right-hand matrix; otherwise, it is not possible to multiply the matrices.

Here is the procedure to find the matrix product:

1. Start with the first

rowofAand the firstcolumnofB. (Note that both of these will containsnelements.)2. Multiply the corresponding elements of this row ad this column.

3. Add all the products together. The result will be the element in the first row and first column of the product matrix

C (AB).4. Repeat this process to find each element of

AB, using theith rowAand thejth column ofBto get the element in theith row and thejth column ofAB. (the resulting matrix will havemrows andpcolumns.)For example:

Multiplying a 3-by-3 matrix A times a 3-by-1 matrix B:

A X B[3 7 11] X [20]

[9 4 8] X [40]

[5 6 10] X [30]

We started by taking the first row of

Aand the first (in this case, only) column ofB.3 x 20 + 7 x 40 + 11 x 30 = 670

The result of this calculation was the element in row 1, column 1 of the product matrix

C (AB). To find the element in row 2, column 1 of the product matrix, we needed to take the second row ofAand the first column ofB:9 x 20 + 4 x 40 + 8 x 30 = 580

Then we needed to that the third row of

Aand the first column ofB:5 x 20 + 6 x 40 + 10 x 30 = 640

Therefore we could write the complete product matrix:

C(AB)=

[3 7 11] X [20] [670]

[9 4 8] X [40] = [580]

[5 6 10] X [30] [640]

Donald Taylor and Sadia Akhtar

Guidelines for Multiplying Matrices

When multiplying matrices you must multiply the same number of rows in the first matrix by the same number of columns in the second. The insides of the equation much match. For example a 2x1 matrix by a 1x1 matrix will work because you have 1 row in the first matrix and 1 column in the second. If you try to multiply a 1x1 matrix by a 2x1 matrix it won't work because you have 1 row in the first matrix and 2 columns in the second. They must match. This is also why matrices aren't commutative because if you flip them they may not work. You also can't multiply a 2 by 1 matrix with a 1 by 2 matrix, but you can multiply a 1 by 2 matrix with a 2 by 1 matrix.

Robert Yemola

Warm Up From 4-1-08

1. Is matrix multiplication commutative? Prove using examples.

No they are not because in the example the below the products are not equal.

1 2 * 5 6 = 19 22

3 4 * 7 8 = 43 50

5 6 * 1 2 = 23 34

7 8 * 3 4 = 31 46

Joshua Hendarto

To determine the answer of the multiplied matrix, its always the outside column times the number of rows

Ex:

Matrix A = 3 by 4

Matrix B = 4 by 6

Answer = 3 by 6

Melanie Thomas**When multiplying matrices, the insides must match up. For example: you can multiply a 1x2 matrix by a 2x1 matrix and the result will be a matrix with the outside values. Example: The result would be a 1x1 matrix.

A matrix is a rectangular array of numbers or algebraic expressions enclosed in square brackets. They are denoted by a capital letter. There are rows and columns. If matrix m has 2 rows 3 columns, the dimensions of m are 2x3. M2,1 is the entry in the second row, first column. m r,c is the rth row and the cth column.

Michael Miller

Matrices are used to describe linear equations of coefficients of linear transformations and to record data.

Milana Lewis-ZakutoA matrix is a an algebraic expression enclosed in square brackets (Some math texts use parentheses instead.) Usually a matrix has rows and columns and in mathematics a row is horizontal and a column is vertical.s

Dakota Townsend

Matrix: An arrangement of data in rows and columns and enclosed brackets[ ].

Matrix Equality: Two matrices are equal when their dimensions are the same and their corresponding entries are equal.

Adding and Subtracting Matrices:

If you have two matrices A and B and if they have the same dimensions than if you add them together the contents inside are the sum of the numbers. Also the dimensions will remain the same.

Example-

1.

2.

Matrices that Represent Systems of Equations:

Sometimes you may get a problem that has 3 different equations in it, and instead of trying to solve them all the regular way you can use a matrix to make it easier.

Example-

Now you can take these equations and make it into a matrix by copying the numbers into the brackets.

A- coefficients

X- variables

B- constants

Multiplying Matrices:

When you multiply matrices you end up with one that is called the product matrix. For the matrix to work the number of columns in the 1rst matrix must be equal to the number of rows in the 2nd matrix.

Now when you do this you have to multiply first than add.

For example-

3.

4

.Inverse of a Matrix:

When finding the inverse of a matrix you must know the identity of it. Usually the identity of a matrix depends on the dimensions.If you have a 3x3 matrix the identity matrix will also be a 3x3 but it will look like this:

Also when making the identity matrix all the 1's go diagonal while the 0's go everywhere else. Basically your goal is to get the identity matrix on the left side.

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