Applications with Matrices Skill 24 Objectives Find determinants
Applications with Matrices Skill 24
Objectives • • Find determinants of square matrices Find inverses of square matrices Use Cramer’s Rule to solve systems of linear equations Use matrices to encode and decode messages
Determinant • Every square matrix can be associated with a real number called its determinant. Determinants have many uses, and several will be discussed in this section. • Historically, the use of determinants arose from special number patterns that occur when systems of linear equations are solved.
Determinant
Example; Determinant of 2 x 2 Matrices
Example Determinants of 3 x 3 Matrix
Inverse of a Matrix The definition of the multiplicative inverse of a matrix is similar to the multiplicative inverse of a number.
Finding the Inverse of a Matrix When a matrix A has an inverse, A is called invertible (or nonsingular); otherwise, A is called singular. A nonsquare matrix cannot have an inverse. To see this, note that if A is of dimension m n and B is of dimension n m (where m n ), then the products AB and BA are of different dimensions and so cannot be equal to each other. Not all square matrices have inverses, as you will see later in this section. When a matrix does have an inverse, however, that inverse is unique.
Inverse of a 2 x 2 Matrix We can use the determinant to find the inverse of a 2 x 2 matrix.
Inverse of Any Square Matrix
Example; Find the Inverse of a 2 x 2 Matrix No Inverse
Example; Find the Inverse of a 3 x 3 Matrix
Example; Find the Inverse of a 3 x 3 Matrix
Example; Find the Inverse of a 3 x 3 Matrix
Cryptography A cryptogram is a message written according to a secret code. (The Greek word kryptos means “hidden. ”) Matrix multiplication can be used to encode and decode messages. To begin, you need to assign a number to each letter in the alphabet (with 0 assigned to a blank space), as follows.
Cryptography - Then the message is converted to numbers and partitioned into uncoded row matrices, each having n entries - To encode a message, choose an n n invertible matrix - Multiply the uncoded row matrices by A (on the right) to obtain coded row matrices.
Example; Encode the Message Encode the message: MEET ME MONDAY - Write the uncoded row matrices of dimension 1 3 for the message To encode a message, use the matrix A.
Example; Encode the Message Encode the message: MEET ME MONDAY Encoded message: 13 -26 21 33 -53 -12 18 -23 -42 5 -20 56 -24 23 77
Example; Decode the Message Decode the message using the inverse of matrix A. 42 88 101 88 201 251 30 33 0 26 56 64 Find the inverse of A. Multiply coded message by inverse of A.
Example; Decode the Message Decode the message using the inverse of matrix A. [42 88 101] [88 201 251] [30 33 0] [26 56 64] [5 14 5] [13 20 0] [6 15 21] [14 4 0] E N E M Y __ F O U N D __
24: Applications with Matrices • Summarize your notes • Questions? • Homework • Worksheet • Quiz
Cramer’s Rule • So far, you have studied three methods for solving a system of linear equations: substitution, elimination with equations, and elimination with matrices. You will now study one more method, Cramer’s Rule, named after Gabriel Cramer (1704– 1752). • This rule uses determinants to write the solution of a system of linear equations.
Example; Matrix Multiplication
Example; Solve Using Cramer’s Rule
Example; Solve Using Cramer’s Rule
Example; Solve Using Cramer’s Rule
Example; Solve Using Cramer’s Rule
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