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Understanding Matrices: Operations, Dimensions, and Inverses

This unit delves into matrices, covering essential concepts including matrix dimensions, elements, and operations such as addition, subtraction, and multiplication. Learn about special types of matrices like zero and equal matrices, as well as scalar multiplication. Explore the determinant of 2x2 and 3x3 matrices and the conditions for a matrix to have an inverse. Understand how to find and use inverses in matrix equations. This comprehensive guide prepares you to tackle problems involving matrices with confidence.

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Understanding Matrices: Operations, Dimensions, and Inverses

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  1. Unit 3: Matrices

  2. Matrices • Matrix: A rectangular arrangement of data into rows and columns, identified by capital letters.

  3. 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)

  4. Determine the dimensions of each matrix.

  5. Elements Matrix Element: Each number in a matrix, identified by its row and column. Example: amn Refers to the m-throw and n-thcolumn

  6. Example Identify each element. • a23 • a12 • a31 • a21

  7. 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.

  8. Adding and Subtracting Matrices • When matrices have the same dimension you add and subtract them by adding or subtracting each corresponding element.

  9. Add or Subtract the following matrices:

  10. A + B • C – B • 2(B + C)

  11. 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.

  12. Using Scalar Products

  13. 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.

  14. Multiplying Matrices Can the following Matrices be multiplied? If so, what dimensions will the product be?? 1. x

  15. Multiplying Matrices Can the following Matrices be multiplied? If so, what dimensions will the product be?? 1. x

  16. Can the following Matrices be multiplied? If so, what dimensions will the product be? 1. 2. 3. 4.

  17. 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.

  18. Matrix Multiplication

  19. You Try!

  20. You Try!

  21. Solutions:

  22. Equivalent Matrices

  23. DETERMINANT of Matrices

  24. Determinant of 2 x 2

  25. Determinant of 2 x 2 • Find the determinant of the following 2x2 matrices:

  26. Determinant of 2 x 2 • Find the determinant of the following 2x2 matrices:

  27. Determinant of a 3x3 Matrix • Step 1: Re-write the first two columns on the right side of the determinant.

  28. 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

  29. 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

  30. 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

  31. First, rewrite the first two columns along side the determinant. Ex. 2: Evaluate using diagonals.

  32. 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.

  33. Example • Find the determinant of the following.

  34. Try Some! • Find the determinant of the following.

  35. Determinant of a 3x3 Matrix

  36. Determinant of a 3x3 Matrix

  37. Inverse of Matrices

  38. Inverse REMEMBER we denote inverse with a -1 power So the inverse of matrix A is A-1

  39. 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.

  40. Inverse exist?! Does the inverse exist?!?!

  41. 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

  42. Determine if the following matrices are inverses. 1. 2.

  43. Finding the Inverse of a 2x2 IF THEN

  44. Find the inverse of the following matrix.

  45. Use your calculator! • 2nd Matrix  Edit • Put in your matrix • 2nd  Matrix  NAME • Get your matrix • X-1

  46. 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

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