Algebra 2 Chapter 4 Notes Matrices & Determinants

Algebra 2 Chapter 4 Notes Matrices & Determinants

4.1 Matrix Operations a MATRIX is a rectangular arrangement of rows & columns DIMENSIONS are the number of rows (horizontal) by the number of columns (vertical) Matrix A = This matrix has 2 rows x 3 columns, so it is a 2 x 3matrix. To add or subtract matrices, just add or subtract the corresponding entries. NOTE: you can add or subtract only if each matrix has the same dimensions. + = - = REMEMBER: “Rows x Columns” when describing a Matrix.

4.1 Multiplying Scalars Multiplying Scalars In matrix algebra, a real number is often called a scalar To multiply a matrix by a scalar, you multiply each entry in the matrix by the scalar. This process is called “Scalar Multiplication.”” = 3 + = + = ─2

4.2 Properties of Matrix Operations REMEMBER: “Rows x Columns” when describing a Matrix. It is NOT “ Columns x Rows”

4.2 Multiplying Matrices The product of 2 matrices A and B is defined provided the number of columns in matrix A = the number of rows in matrix B A x B = AB m x nn x p m x p dimensions of AB Example 1: 3 x 2 x 2 x 4 = 3 x 4 6 entries 8 entries 12 entries

4.2 Properties of Matrix Multiplications To multiply each matrix, multiply EACH Row by EACH column

4.2 Multiplying Matrices A B AB 3 x 2x 2 x 2 = 3 x 2 • = = = REMEMBER: you multiply each ENTIRE row by each ENTIRE column. You multiple and then add the result. That is why the number of columns of the first matrix must equal the number of rows of the second matrix.

4.2 Multiplying Matrices A B AB 3 x 2x 2 x 2 = 3 x 2 • = = = REMEMBER: you multiply each ENTIRE row by each ENTIRE column and then add the result. That is why the number of columns of the first matrix must equal the number of rows of the second matrix.

4.2 Multiplying Matrices A B AB 3 x 2x 2 x 2 = 3 x 2 • = = = REMEMBER: you multiply each row by each column. That is why the number of columns of the first matrix must equal the number of rows of the second matrix.

4.3 Determinants and Cramer’s Rule Determinants and Cramer’s Rule Associated with each square matrix is a real # called it’s determinant, denoted by det A or │ A │. The determinant of a 2 x 2 matrix is the difference of the products of the entries on the diagonals det = = ad ─bc Example 1 det = = 1 • 5 ─ 2 • 3 = 5 ─ 6 = ─ 1

4.3 Determinants and Cramer’s Rule 2 steps to get the determinant of a 3 x 3 matrix: Repeat first 2 columns to the right of the matrix Select the sum of the products the products in red from the sum of the products in blue. det (aei+ bfg + cdh) ─ (gec + hfa + idb) = = Example 1 = (0 + ─ 1 + ─ 12) ─ (0 + 4 + 8) = ─ 13 ─12 = det = ─ 25

4.3 Determinants and Cramer’s Rule A Determinant can find the area of a triangle whose vertices are points in a coordinate plane. Area of a Triangle with vertices , ( x1 , y1 ) , ( x2 , y2 ) , ( x3 , y3 ) ( x1 , y1 ) • A = ± ½ • • ( x2 , y2 ) ( x3 , y3 ) Where ± indicates the appropriate sign should be chosen to yield a positive sign. Example 1 A = ± ½ = ± ½ (0 + 12 + 8) ─ (0 + 2 + 8) = ½ (20 ─10) = 5

4.3 Determinants and Cramer’s Rule Cramer’s Rule for a 2 x 2 matrix: Let A be the coefficient matrix of this linear system. Linear System: ax + by = e cx + dy = f Coefficient Matrix: = det A If det A ≠ 0, then the system has exactly one solution, the solution is: x = y = and det A det A

4.3 Determinants and Cramer’s Rule Cramer’s Rule for a 2 x 2 matrix: Let A be the coefficient matrix of this linear system. 8 x + 5 y = 2 2 x ─4 y = ─10 Coefficient Matrix: Linear System: a x + b y = e c x + d y = f (8) (─ 4 ) ─ (2) (5) = (─ 32 ) ─ (10) det A (─ 42 ) = If det A ≠ 0, then the system has exactly one solution, the solution is: = = = (─ 8) +(50) = 42 = x = (2) (─ 4 ) ─ (─10) (5) ─ 1 ─ 42 ─ 42 ─ 42 det A ─ 42 (8) (─ 10) ─ (2) (2) = 2 = (─ 80) ─ (4) ─ 84 = = = y = ─ 42 ─ 42 ─ 42 det A ─ 42 ( x , y ) (─ 1 , 2 )

4.3 Cramer’s Rule for a 3 x 3 matrix: Let A be the coefficient matrix of this linear system. Linear System: a x + b y + c z = j d x + e y + f z = k g x + h y + iz = l Coefficient Matrix: = ( aei +bfg + cdh) – ( ceg + afh + bdi) = det A If det A ≠ 0, then the system has exactly one solution, the solution is: x = det A y = det A ( x , y , z ) z = det A

4.3 Cramer’s Rule for a 3 x 3 matrix: Let A be the coefficient matrix of this linear system. Linear System: x + 4 y = 16 3 x + 8 y + 3 z = 92 2 y + z = 18 Coefficient Matrix: = ( 8 + 0 + 0 ) – ( 0 + 6 + 12 ) = – 10 = det A If det A ≠ 0, then the system has exactly one solution, the solution is: = ( 128 + 216 + 0 ) – ( 0 + 96 + 368) – 10 = –120 – 10 = 12 x = ( x , y , z ) ( 12 , 1 , 16 ) – 10 = –10 – 10 = 1 = ( 92 + 0 + 0 ) – ( 0 + 54 + 48) – 10 y = – 10 = ( 144 + 0 + 96 ) – ( 0 + 184 + 216) – 10 = –160 – 10 = 16 z = – 10

4.4 Identity and Inverse Matricies Identity and Inverse Matrices The number 1 is the multiplicative identity for real #’s because 1 • a = a • 1 = a For matrices, the n x n Identity matrix is the matrix that has 1’s on the main diagonal and 0’s elsewhere. 3 x 3 Identity matrix 2 x 2 Identity matrix I = I = If A is any n x n matrix and I is the Identity matrix, then I A = A and A I = A Two n x n matrices are inverses of each other if their products (in both orders) is the n x n identity matrix. Remember: 3/4 • 4/3 = 1 AB = = = I BA = = = I

4.4 Identity and Inverse Matricies Inverse Matrices The inverse of a 2 x 2 matrix A = is A –1 = 1 │ A │ 1 ad–cb = provided ad–cb ≠ 0 Remember : │ A │ andad–cbare the determinant of matrix A. Example: A = 1 3•2–4•1 1 6–4 1 2 A –1 = = = =

Inverse Matrices 4.4 A = • Switch a and d • Change b and c to opposite signs • Multiply by 1 divided by determinant of matrix A 1 ad–cb A –1 =

4.4 Solving a Matrix Equation A B Example 2: Solve matrix equation, A X = B for 2 x 2 matrix X X = A X = B 1 A X = 1 B A A 1 X = B A X = B A Solution: Find inverse of A A –1 = 1 4 – 3 = X = X = X =

4.4 Solving Systems using Inverse Matricies Solving systems using inverse matricies Coefficient matrix Matrix of variables Matrix of constants 5 x– 4 y = 8 1 x + 2 y = 6 Remember: Row x Column • = Example 1: Write a system of linear equations as a matrix equation – 3 x+ 4 y = 5 2 x – 1 y = – 10 = Example 2: Use matricies to solve the linear system = = A –1 = 1 3 – 8 = A –1 = 1 – 5 = ( X , Y ) (─ 7 , – 4 )

Algebra 2 Chapter 4 Notes Matrices & Determinants