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Unit 3: Matrices. Matrices. Matrix : A rectangular arrangement of data into rows and columns, identified by capital letters . . Dimensions. Matrix Dimensions: Number of rows, m, by the number of columns, n. Read as “m by n” matrix . Also known as the order of a matrix .
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Matrices • Matrix: A rectangular arrangement of data into rows and columns, identified by capital letters.
Dimensions • Matrix Dimensions: Number of rows, m, by the number of columns, n. Read as “m by n” matrix. Also known as the order of a matrix. • RBC (ROWS BY COLUMNS)
Elements Matrix Element: Each number in a matrix, identified by its row and column. Example: amn Refers to the m-throw and n-thcolumn
Example Identify each element. • a23 • a12 • a31 • a21
Zero Matrix: The additive IDENTITY of matrices. A matrix whose elements are all zeros. • Equal Matrices: Matrices with the same dimensions and equal corresponding elements. • Scalar: A real number factor.
Adding and Subtracting Matrices • When matrices have the same dimension you add and subtract them by adding or subtracting each corresponding element.
A + B • C – B • 2(B + C)
A matrix equation is an equation in which the variable is a matrix. You can solve for the variable by adding or subtracting a matrix a matrix to both sides to an equation.
Matrix Multiplication • When multiplying matrices A and B, the number of COLUMNS in matrix A MUST be equal to the ROWS in matrix B. The size of the product is: # rows in A x # columns in B.
Multiplying Matrices Can the following Matrices be multiplied? If so, what dimensions will the product be?? 1. x
Multiplying Matrices Can the following Matrices be multiplied? If so, what dimensions will the product be?? 1. x
Can the following Matrices be multiplied? If so, what dimensions will the product be? 1. 2. 3. 4.
How to multiply matrices • Multiply the elements of each row in the first matrix by the elements in each column of the second matrix • Add the products to get the new element.
Determinant of 2 x 2 • Find the determinant of the following 2x2 matrices:
Determinant of 2 x 2 • Find the determinant of the following 2x2 matrices:
Determinant of a 3x3 Matrix • Step 1: Re-write the first two columns on the right side of the determinant.
STEP 2: Draw a diagonal from each element in the top row diagonally downward. Find the product of the numbers on each diagonal. aei bfg cdh
STEP 3: Then draw a diagonal from each element in the bottom row diagonally upward. Find the product of the numbers on each . gec hfa idb
Step 4: Add the products in the first set of diagonals, and then subtract the products from the second set of diagonals. The value is: • aei + bfg + cdh– (gec + hfa + idb) • or aei + bfg + cdh– gec – hfa – idb
First, rewrite the first two columns along side the determinant. Ex. 2: Evaluate using diagonals.
Next, find the values using the diagonals. -5 0 24 Ex. 2: Evaluate using diagonals. 0 4 60 Now add the bottom products and subtract the top products. 4 + 60 + 0– 0 – (-5) – 24 = 45. The value of the determinant is 45.
Example • Find the determinant of the following.
Try Some! • Find the determinant of the following.
Inverse REMEMBER we denote inverse with a -1 power So the inverse of matrix A is A-1
Requirement to have an Inverse Matrix MUST be square, meaning it has the same number of rows and columns Matrix MUST NOT have a determinant of zero.
Inverse exist?! Does the inverse exist?!?!
Multiplying Inverse When you Multiply a matrix A times it’s inverse, the Product is the Identity Matrix. Identity Matrix is a square matrix where the top left to Bottomright diagonal are all ones, and everything else is a zero
Finding the Inverse of a 2x2 IF THEN
Use your calculator! • 2nd Matrix Edit • Put in your matrix • 2nd Matrix NAME • Get your matrix • X-1
The inverse of a matrix can be usedwhen solving matrix equations. For Matrices A and B, we can find Matrix X: IF AX = B THEN X = A-1B
