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H23-1 - Math 330 Homework 2.3(Pages 132,133 In all...

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Math 330 Homework 2.3 (Pages 132,133) In all exercises unless stated otherwise, the matrices indicated are square (i.e. as many rows as columns). (6) Use as few steps as possible to decide if the following matrix is invertible: 1 - 5 - 4 0 3 4 - 3 6 0 SOLUTION: The key to checking if the matrix is invertible, is to make sure that every column is a pivot column. Seeing whether or not this is the case, simply involves moving to Echelon (not reduced Echelon) form. Since we are not attempting to find the inverse, we don’t need to augment by the identity: R 3 R 3+3 R 1 = 1 - 5 - 4 0 3 4 0 - 9 - 12 R 3 R 3+3 R 2 = 1 - 5 - 4 0 3 4 0 0 0 Therefore, we only have two pivots so this matrix is not invertible. (8) Use as few steps as possible to decide if the following matrix is invertible: 1 3 7 4 0 5 9 6 0 0 2 8 0 0 2 10 SOLUTION: R 4 R 4 - R 3 = 1 3 7 4 0 5 9 6 0 0 2 8 0 0 0 2 Now we have a matrix in Echelon form with a pivot in every column. Hence this matrix is invertible. (12) TRUE or FALSE The student should be reminded that all matrices are assumed to be square. The validity of these statements is very different if we do not assume this. (a) If there is an n × n matrix D such that AD = I , then there is also an n × n matrix C such that AC = I .
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