Column 1 Row 2 Row 3 Row m
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Column 1 Row 2 Row 3 Row m Column 2 Column 3 Column 4
A matrix of m rows and n columns is called a matrix with dimensions m x n. 2 X 3 2 X 1 3 X 3 1 X 2
3 X 2 1 X 2 2 X 1 3 X 3 1 X 1
If A and B are both m × n matrices then the sum of A and B, denoted A + B, is a matrix obtained by adding corresponding elements of AA and B. B. add these add these
When a zero matrix is added to another matrix of the same dimension, that same matrix is obtained.
To subtract matrices, we subtract the corresponding elements. The matrices must have the same dimensions.
If A is an m × n matrix and s is a scalar, then we let k. A denote the matrix obtained by multiplying every element of A by k. This procedure is called scalar multiplication. PROPERTIES OF SCALAR MULTIPLICATION
The m × n zero matrix, denoted 0, is the m × n matrix whose elements are all zeros. 2× 2 1× 3
Scalar Multiplication:
6 x+8=26 6 x=18 x=3 10 -2 y=8 -2 y=-2 y=1
The multiplication of matrices is easier shown than put into words. You multiply the rows of the first matrix with the columns of the second adding products Find AB First we multiply across the first row and down the first column adding products. We put the answer in the first row, first column of the answer.
Find AB Notice the sizes of A and B and the size of the product AB. Now we multiply across the first second rowrow andand down the second first We multiplied across first row and down first column second and we’lland putwe’ll the answer put the in answer the second first in the row, second firstrow, so we put the answer in the first row, first column. second
To multiply matrices A and B look at their dimensions MUST BE SAME SIZE OF PRODUCT If the number of columns of A does not equal the number of rows of B then the product AB is undefined.
Now let’s look at the product BA. Commuter's Beware! Completely different than AB!
PROPERTIES OF MATRIX MULTIPLICATION Is it possible for AB = BA ? Yes it is possible.
The Identity Matrix for Multiplication Let A be a square matrix with n rows and n columns. Let I be a matrix with the same dimensions and with 1’s on the main diagonal and 0’s elsewhere. Then AI = IA = A
What is AI? Multiplying a matrix by the identity gives the matrix back again. What is IA? an n n matrix with ones on the main diagonal and zeros elsewhere
The Inverse of a Matrix Let A be a square matrix with n rows and n columns. If there is an n x n matrix B such that AB = I and BA = I, then A and B are inverses of one another. The inverse of matrix A is denoted by A-1.
Can we find a matrix to multiply the first matrix by to get the identity? ? Let A be an n n matrix. If there exists a matrix B such that AB = BA = I then we call this matrix the inverse of A and denote it A-1.
If A has an inverse we say that A is nonsingular. A-1 does not exist we say A is singular. If To matrix A, A, a a To find the inverse of of a a matrix we we put the matrix line identity matrix. We We then perform row line and then the identity operations toturnititintothe theidentity. We We operations on on matrix A A to carry right hand side carry the row operations across and the right will turn into the inverse. r 2 r 1 r 2 2 r 1+r 2
Check this answer by multiplying. We should get the identity matrix if we’ve found the inverse.
Cofactor Method for Inverses • Let A = (aij) be an nxn matrix • Recall, the co-factor Cij of element aij is: Cij = (-1)i+j |Mij| • Mij is the (n-1) x (n-1) matrix made by removing the ROW i and COLUMN j of A
Cofactor Method for Inverses • Put all co-factors in a matrix – called the matrix of co-factors: C 11 C 12 C 21 C 22 C 1 n C 2 n Cn 1 Cn 2 Cnn
Cofactor Method for Inverses • Inverse of A is given by: A-1 1 = (matrix of co-factors)T |A| 1 = |A| C 11 C 21 C 12 C 22 Cn 1 Cn 2 C 1 n C 2 n Cnn
Examples • Calculate the inverse of A = M 11 = d |M 11| = d a b c d C 11 = d
Examples • Calculate the inverse of A = M 12 = c |M 12| = c a b c d C 12 = -c
Examples • Calculate the inverse of A = M 21 = b |M 21| = b a b c d C 12 = -b
Examples • Calculate the inverse of A = M 22 = a |M 22| = a a b c d C 22 = a
Examples • Calculate the inverse of A = a b c d • Found that: C 11 = d C 12 = -c C 21 = -b C 22 = a • So, A-1 1 = (matrix of co-factors)T |A|
Examples • Calculate the inverse of A = a b c d • Found that: C 11 = d C 12 = -c C 21 = -b C 22 = a • So, A-1 1 = (matrix of co-factors)T (ad-bc)
Examples • Calculate the inverse of A = a b c d • Found that: C 11 = d C 12 = -c C 21 = -b C 22 = a • So, C 11 C 12 1 A-1 = (ad-bc) C C 21 22 T
Examples • Calculate the inverse of A = a b c d • Found that: C 11 = d C 12 = -c C 21 = -b C 22 = a • So, C 11 C 21 1 A-1 = (ad-bc) C C 12 22
Examples • Calculate the inverse of A = a b c d • Found that: C 11 = d C 12 = -c C 21 = -b C 22 = a • So, C 21 d 1 A-1 = (ad-bc) C C 12 22
Examples • Calculate the inverse of A = a b c d • Found that: C 11 = d C 12 = -c C 21 = -b C 22 = a • So, A-1 d -b 1 = (ad-bc) C C 12 22
Examples • Calculate the inverse of A = a b c d • Found that: C 11 = d C 12 = -c C 21 = -b C 22 = a • So, A-1 d -b 1 = (ad-bc) -c C 22
Examples • Calculate the inverse of A = a b c d • Found that: C 11 = d C 12 = -c C 21 = -b C 22 = a • So, A-1 d -b 1 = (ad-bc) -c a
Examples 3 x 3 Matrix 1 • Calculate the inverse of B = 1 2 1 1 2 2 3 4 • Find the co-factors: 2 2 M 11 = 3 4 |M 11| = 2 C 11 = 2
Examples 3 x 3 Matrix 1 • Calculate the inverse of B = 1 2 1 1 2 2 3 4 • Find the co-factors: 1 2 M 12 = 2 4 |M 12| = 0 C 12 = 0
Examples 3 x 3 Matrix 1 • Calculate the inverse of B = 1 2 1 1 2 2 3 4 • Find the co-factors: 1 2 M 13 = 2 3 |M 13| = -1 C 13 = -1
Examples 3 x 3 Matrix 1 • Calculate the inverse of B = 1 2 1 1 2 2 3 4 • Find the co-factors: 1 1 M 21 = 3 4 |M 21| = 1 C 21 = -1
Examples 3 x 3 Matrix 1 • Calculate the inverse of B = 1 2 1 1 2 2 3 4 • Find the co-factors: 1 1 M 22 = 2 4 |M 22| = 2 C 22 = 2
Examples 3 x 3 Matrix 1 • Calculate the inverse of B = 1 2 1 1 2 2 3 4 • Find the co-factors: 1 1 M 23 = 2 3 |M 23| = 1 C 23 = -1
Examples 3 x 3 Matrix 1 • Calculate the inverse of B = 1 2 1 1 2 2 3 4 • Find the co-factors: 1 1 M 31 = 2 2 |M 31| = 0 C 31 = 0
Examples 3 x 3 Matrix 1 • Calculate the inverse of B = 1 2 1 1 2 2 3 4 • Find the co-factors: 1 1 M 32 = 1 2 |M 32| = 1 C 32 = -1
Examples 3 x 3 Matrix 1 • Calculate the inverse of B = 1 2 1 1 2 2 3 4 • First find the co-factors: 1 1 M 33 = 1 2 |M 33| = 1 C 33 = 1
Examples 3 x 3 Matrix 1 • Calculate the inverse of B = 1 2 1 1 2 2 3 4 • Next the determinant: use the top row: |B| = 1 x |M 11| -1 x |M 12| + 1 x |M 13| = 2 – 0 + (-1) = 1
Examples 3 x 3 Matrix • Using the formula, B-1 1 = (matrix of co-factors)T |B| 1 = (matrix of co-factors)T 1
Examples 3 x 3 Matrix • Using the formula, B-1 1 = (matrix of co-factors)T |B| 1 2 0 1 = 1 -1 2 -1 0 -1 1 T
Examples 3 x 3 Matrix • Using the formula, B-1 1 = (matrix of co-factors)T |B| = 2 -1 0 0 2 -1 -1 -1 1 • Same answer obtained by Gauss-Jordan method
We can use A-1 to solve a system of equations To see how, we can re-write a system of equations as matrices. coefficient matrix variable matrix constant matrix
left multiply both sides by the inverse of A This is just the identity but the identity times a matrix just gives us back the matrix so we This then gives us a formula have: for finding the variable matrix: Multiply A inverse by the constants.
find the inverse -2 r 1+r 2 r 1 -3 r 2 -r 2 x y This is the answer to the system
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