Explanation Given Information: The provided matrix is [ 1 2 3 4 1 3 4 2 0 1 2 3 0 0 1 2 ] . Formula used: A matrix with m rows and n columns is of dimension m × n , where m and n are positive integers. Two matrices A and B are inverse pairs if A B = I , where I is the n × n identity matrix. To obtain inverse of a n × n matrix A , augment the matrix with n × n identity matrix and row reduce the obtained matrix. If the matrix obtained after row reduction has identity matrix on the left side then the matrix obtained on the right is the required inverse of A otherwise A is singular. Calculation: Consider the matrix [ 1 2 3 4 1 3 4 2 0 1 2 3 0 0 1 2 ] . Recall that a matrix with m rows and n columns is of dimension m × n , where m and n are positive integers. Since, number of rows in [ 1 2 3 4 1 3 4 2 0 1 2 3 0 0 1 2 ] is 4 and number of columns is 4 , Substitute 4 for m and 4 for n in m × n . Thus, dimension of [ 1 2 3 4 1 3 4 2 0 1 2 3 0 0 1 2 ] is 4 × 4 . Augment the matrix with the 4 × 4 identity matrix [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] . Thus, the augmented matrix is [ 1 2 3 4 1 0 0 0 1 3 4 2 0 1 0 0 0 1 2 3 0 0 1 0 0 0 1 2 0 0 0 1 ] . Row reduce the matrix [ 1 2 3 4 1 0 0 0 1 3 4 2 0 1 0 0 0 1 2 3 0 0 1 0 0 0 1 2 0 0 0 1 ] . Perform the row operations R 2 − R 1 . [ 1 2 3 4 1 0 0 0 1 3 4 2 0 1 0 0 0 1 2 3 0 0 1 0 0 0 1 2 0 0 0 1 ] → [ 1 2 3 4 1 0 0 0 0 1 1 − 2 − 1 1 0 0 0 1 2 3 0 0 1 0 0 0 1 2 0 0 0 1 ] Perform the row operations R 1 − 2 R 2 and R 3 − R 2 . [ 1 2 3 4 1 0 0 0 0 1 1 − 2 − 1 1 0 0 0 1 2 3 0 0 1 0 0 0 1 2 0 0 0 1 ] → [ 1 0 1 8 3 − 2 0 0 0 1 1 − 2 − 1 1 0 0 0 0 1 5 1 − 1 1 0 0 0 1 2 0 0 0 1 ] Perform the row operations R 1 − R 3 , R 2 − R 3 and R 4 − R 3 . [ 1 0 1 8 3 − 2 0 0 0 1 1 − 2 − 1 1 0 0 0 0 1 5 1 − 1 1 0 0 0 1 2 0 0 0 1 ] → [ 1 0 0 3 2 − 1 − 1 0 0 1 1 − 2 − 2 2 − 1 0 0 0 1 5 1 − 1 1 0 0 0 0 − 3 − 1 1 − 1 1 ] Perform the row operations R 1 + R 4 , 3 R 2 − 7 R 4 and 3 R 3 + 5 R 4

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ISBN: 9781337291262

Author: Stefan Waner, Steven Costenoble

Publisher: Brooks Cole

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Chapter 4, Problem 14RE

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Ch. 4.1 - In Exercises 110, find the dimensions of the given...Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - In Exercises 110, find the dimensions of the given...Ch. 4.1 - In Exercises 110, find the dimensions of the given...Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - In Exercises 110, find the dimensions of the given...

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The inverse of the matrix [ 1 2 3 4 1 3 4 2 0 1 2 3 0 0 1 2 ] .

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Chapter 4 Solutions