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4.5 Matrices, Determinants, Inverseres
-Identity matrices-Inverse matrix (intro)-An application-Finding inverse matrices (by hand)-Finding inverse matrices (using calculator)
A review of the Identity
• For real numbers, what is the additive identity?
• Zero…. Why?
• Because for any real number b, 0 + b = b
• What is the multiplicative identity?
• 1 … Why?
• Because for any real number b, 1 * b = b
Identity Matrices
• The identity matrix is a square matrix (same # of rows and columns) that, when multiplied by another matrix, equals that same matrix
• If A is any n x n matrix and I is the n x n Identity matrix, then A * I = A and I*A = A
Examples
• The 2 x 2 Identity matrix is:
• The 3 x 3 Identity matrix is:
1 0
0 1
1 0 0
0 1 0
0 0 1
•Notice any pattern?
•Most of the elements are 0, except those in the diagonal from upper left to lower right, in which every element is 1!
Inverse review
• Recall that we defined the inverse of a real number b to be a real number a such that a and b combined to form the identity
• For example, 3 and -3 are additive inverses since 3 + -3 = 0, the additive identity
• Also, -2 and – ½ are multiplicative inverses since (-2) *(- ½ ) = 1, the multiplicative identity
Matrix Inverses
• Two n x n matrices are inverses of each other if their product is the identity
• Not all matrices have inverses (more on this later)
• Often we symbolize the inverse of a matrix by writing it with an exponent of (-1)
• For example, the inverse of matrix A is A-1
• A * A-1 = I, the identity matrix.. Also A-1 *A = I• To determine if 2 matrices are inverses, multiply
them and see if the result is the Identity matrix!
Determine whether X and Y are inverses.
Determine whether P and Q are inverses.
Determine whether each pair of matrices are inverses.
a.
b.
How do we find the inverse???
• 1st find the determinant• The determinant;
– determines whether the inverse of a matrix exists.– influences the elements the inverse contains
• For the matrix shown below, the determinant is equal to ad – bc
• In words, multiply the elements in each diagonal, then subtract the products!
• Order Matters.a b
c d
More about determinants
• If the determinant of a matrix equals zero, then the inverse of that matrix does not exist!
• Every square matrix has a determinant. • We will use DETERMINANTS and INVERSES to
solve matrix equations of the type AX = B.
Finding the inverse of a 2 x 2 matrix
• For the matrix:
• The inverse is found by calculating:
a b
c d
1 d b
c aad bc
In other words: -Switch the elements a and d -Reverse the signs of the elements b and c -Multiply by the fraction (1 / determinant)
Find the inverse of the matrix, if it exists.
Find the value of the determinant.
Since the determinant is not equal to 0, S –1 exists.
Find the inverse of each matrix, if it exists.
a.
b.
Answer: No inverse exists.
Answer:
Find the inverse of the matrix, if it exists.
Find the value of the determinant.
Answer: Since the determinant equals 0, T –1 does
not exist.
Definition of inverse
a = –1, b = 0,c = 8, d = –2
Answer: Simplify.
Check: