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Transcript
Matrix Vocabulary Dimensions of a matrix- The dimensions are determined by the number of rows and columns. 2 4 1X2 One row, 2 columns 1 3 0 0 0 1 2X3 2 rows, 3 columns Entries- The numbers inside the matrix. a11 a21 a 31 1 1 1 1 2 2 2 2 3 3 3 3 3X4 3 rows, 4 columns a13 a23 a33 a12 a22 a32 a21 means the entry is in the second row, first column. a34 means the entry is in the third row, fourth column. Square Matrix- There is an equal number of rows and columns a11 a12 2X2 a21 a22 Rectangular Matrix- The number of rows is not equal to the number of columns. a11 a12 a13 2X3 a21 a22 a23 Column vector- One column of entries 1 2 m Row vector- A row of entries. 1 2 mX1 m 1Xm Addition of matrices Definition: a b e c d g f ae b f h c g d h Example: 1 4 3 5 1 (3) 4 (5) 67 8 6 0 7 8 0 2 1 8 13 *In order to add or subtract two matrices, their dimensions must be the same. Subtraction of matrices Definition: a b e c d g f ae b f h c g d h Example: 1 4 3 5 1 (3) 4 (5) 67 8 6 0 7 8 0 4 9 8 1 *In order to add or subtract two matrices, their dimensions must be the same *In order to multiply two matrices, the # of columns of the first matrix must equal the number of rows of the second. To determine if it is possible to multiply two matrices check the number of columns of the first matrix with the rows of the second (The inside numbers): 2 X 5 and 5 X 3 → 4 X 6 and 4 X 6 → Yes No The answer will be determined by the rows of the first matrix and the columns of the second (The outside numbers): 3 X 5 and 5 X 2 → 2 X 4 and 4 X 5 → 3X2 2X5 Multiplication of Matrices Definition: a b e x c d g f ae bg h ce dg af bh cf dh Example: 1 2 3 5 1(3) 2(5) 1(5) 2(1) x 3 4 5 1 3(3) 4(5) 3(5) 4(1) 1 2 3 5 7 7 x 3 4 5 1 11 19 Identity Matrix The identity matrix, I, is similar to the 1 in standard multiplication. Remember the identity property: Anything times 1 is itself. 1 2 1 0 1 2 3 1 0 1 3 1 1 1X1 1 0 0 1 2X2 1 0 0 0 1 0 0 0 1 3X3 AI = A The identity matrix is a matrix consisting of 1’s and 0’s. The ones are found along the diagonal of the matrix starting in the top right corner. The identity matrix is always a square matrix. The size is determined by the number of columns of the original matrix. A 2 X 3 matrix would have a 3 X 3 identity. Inverse Matrix 1 The inverse matrix, A , is the matrix that will produce an identity matrix when multiplied by the original matrix. It can be used to solve systems of equations. Rectangular matrices do not have inverses. Not all square matrices have inverses. If we have a matrix equation Ax B you can solve by multiplying both sides of the equation by A1 , A1 Ax A1 B . Let’s say we have the system of equations: x 2y 5 3 x y 2 1 2 x 5 We could write the matrix equation: 3 1 y 2 1 2 0.2 0.4 1 So A and A and 3 1 0.6 0.2 x 0.2 0.4 5 y 0.6 0.2 2 x 1.8 From this multiplication we get y 3.4