**Dimensions of a Matrix: The number of horizontal rows it has x the number of vertical columns.
*****Dimensions = 1) Addition/Subtraction. 2) Scalar multiplication.
* Matrix multiplication is NOT commutative.
Ex: [3 4 5 6]
dimensions: 1 x 4
Adding Matrices:
*To add matrices, you have to add the corresponding entries (you can only add matrices of the same size)
*To add matrices, you have to add the corresponding entries (you can only add matrices of the same size)
*Row by Column
Ex: [1 2 3] + [2 1 2] = [3 3 5]
[1 0 2] [1 0 3] [2 0 5]
Subtarcting Matrices:
*To subtract matrices, you have to subtract the corresponding matrices.
*Row by Column
Scalar Multiplication: If a matrix is multiplied by a scalar, each element of the matrix is multiplied by that number.
* Multiply every entry by a commen #.
*Row by Column.
Ex: 2A = [1 2 3] > 2 (1) 2 (2) 2 (3) =
[5 0 -1] 2 4 6 = [2 4 6]
2(5) 2(0) 2(-2) = [10 0 -2]
10 0 -2 =
Matrix multiplication:
*Matrix mutiplication involves both multiplication and addition.
*Matrix multiplication is NOT communitive.
*Order matter: communitive property.
Ex 1: [8 6] [-1 7] = [-28 44 ]
[12 10] X [6 -2] = [48 64]
[-3 2] [15 -25]
Ex 2: [5 0] [-1 7] = 5(-1) + 0(6) 5(7) + 0(-2)
[4 7] X [6 -2] = -5+0 35+0
-5 35 = [5 35]
4(-1) + 7(6) 4(7) + 7(-2) = [40 14]
-4 + 42 28 + (-14)
40 14
Identity Matrix for multiplication:
* The identity matrix, I, is a matrix that has all 1's in its diagonal, & 0's everywhere else.
Ex:
The Inverse of a matrix:
* The inverse of matrix A, denoted A^-1, is a matrix such that the product of AA^-1 = I
Ex:
The determinate of a 2x2 Matrix:
* lAl = means det A = ad - bc
A = [1 4] 6(1) - 4(3) = -6
[3 6] 6 - 12 =
* If det (A) = 0, the matrix does NOT have an inverse.
*To subtract matrices, you have to subtract the corresponding matrices.
*Row by Column
Scalar Multiplication: If a matrix is multiplied by a scalar, each element of the matrix is multiplied by that number.
* Multiply every entry by a commen #.
*Row by Column.
Ex: 2A = [1 2 3] > 2 (1) 2 (2) 2 (3) =
[5 0 -1] 2 4 6 = [2 4 6]
2(5) 2(0) 2(-2) = [10 0 -2]
10 0 -2 =
Matrix multiplication:
*Matrix mutiplication involves both multiplication and addition.
*Matrix multiplication is NOT communitive.
*Order matter: communitive property.
Ex 1: [8 6] [-1 7] = [-28 44 ]
[12 10] X [6 -2] = [48 64]
[-3 2] [15 -25]
Ex 2: [5 0] [-1 7] = 5(-1) + 0(6) 5(7) + 0(-2)
[4 7] X [6 -2] = -5+0 35+0
-5 35 = [5 35]
4(-1) + 7(6) 4(7) + 7(-2) = [40 14]
-4 + 42 28 + (-14)
40 14
Identity Matrix for multiplication:
* The identity matrix, I, is a matrix that has all 1's in its diagonal, & 0's everywhere else.
Ex:
The Inverse of a matrix:
* The inverse of matrix A, denoted A^-1, is a matrix such that the product of AA^-1 = I
Ex:
The determinate of a 2x2 Matrix:
* lAl = means det A = ad - bc
A = [1 4] 6(1) - 4(3) = -6
[3 6] 6 - 12 =
* If det (A) = 0, the matrix does NOT have an inverse.
