Copy of tut2.pdf - MA 106 Spring 2014 Tutorial Sheet 2 1 3 −2 1 Find the inverse of the matrix 2 0 5 −7 using elementary row operations −4 1 1

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MA 106: Spring 2014: Tutorial Sheet 2 1. Find the inverse of the matrix 1 3 - 2 2 5 - 7 0 1 - 4 using elementary row operations. 2. Let A = 1 1 1 0 -3 -17 1 2 4 -24 8 -5 0 -7 2 2 . Assuming that A is invertible, find the last row and the last column of A - 1 . 3. Let A be an m × n matrix, E ij be an n × n matrix unit, and E be an n × n elementary matrix. Prove the following. (i) Post-multiplying A by E ij replaces the j th column of A by the i th column by A , and all other columns of A by 0. (ii) If E = I + αE ij is of type I, then post-multiplying A by E adds α times the i th column of A to its j th column. (iii) If E is of type II, then post-multiplying A by E interchanges the i th and the j th columns of A . (iv) If E = I + ( α - 1) E ij is of type III, then post-multiplying A by E multiplies the j th column of A by α . 4. Prove that det a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 a 22 a 33 - a 11 a 32 a 23 - a 21 a 12 a 33 + a 21 a 32 a 13 + a 31 a 12 a 23 - a 31 a
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  • Spring '16
  • Ravi Banavar

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