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WORKSHEET 13 INVERSES OF SQUARE MATRICES In the last ...

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<strong>WORKSHEET</strong> <strong>13</strong><br />

<strong>INVERSES</strong> <strong>OF</strong> <strong>SQUARE</strong> <strong>MATRICES</strong><br />

DAVID MEREDITH<br />

<strong>In</strong> <strong>the</strong> <strong>last</strong> worksheet, you showed that if A is a square matrix, <strong>the</strong>n A has a<br />

left inverse if and only if it has a right inverse if and only if <strong>the</strong> number of rows or<br />

columns is <strong>the</strong> rank of A. <strong>In</strong> fact any one-sided inverse of A is <strong>the</strong> unique two-sided<br />

inverse of A, denoted A −1 .<br />

(1) Show that a square matrix has an inverse if and only if it row reduces to<br />

<strong>the</strong> identity.<br />

(2) <strong>In</strong> Octave construct a 5 × 5 matrix A of rank 5.<br />

(3) Construct B = A −1 with <strong>the</strong> following commands, and show <strong>the</strong> results are<br />

all <strong>the</strong> same:<br />

A^(-1), eye(5)/A, A\eye(5), inverse(A), inv(A)<br />

Check that AB = BA = I.<br />

(4) Construct a column vector b with five entries. Solve <strong>the</strong> equation Ax = b<br />

by calculating x = A −1 b. Show that <strong>the</strong>re are two ways to calculate <strong>the</strong><br />

answer with Octave: x = A^(-1)*b, x = A\b. Check that your solution<br />

is correct.<br />

(5) For 2 × 2 matrices <strong>the</strong>re is a simple formula for <strong>the</strong> inverse, when it exists.<br />

Put Octave aside for([ a moment. ]) a b<br />

(a) Show that rank<br />

= 2 if and only if ad − bc ≠ 0. Argue that<br />

c d<br />

<strong>the</strong> matrix has rank 2 iff its rows are linearly independent iff <strong>the</strong> rows<br />

are not parallel iff ad − bc ≠ 0. Fill in <strong>the</strong> gaps in <strong>the</strong> argument. The<br />

quantity ad − bc is <strong>the</strong> determinant of <strong>the</strong> matrix. (Extra information:<br />

larger square matrices also have determinants, which are 0 precisely<br />

when <strong>the</strong> matrix is not invertible. However <strong>the</strong> formula for <strong>the</strong> determinant<br />

is complex, so it’s best to let Octave calculate determinants of<br />

larger matrices with <strong>the</strong> command det(A).)<br />

[ ] −1 [ ]<br />

a b 1 d −b<br />

(b) Show that =<br />

.<br />

c d<br />

[<br />

ad<br />

]<br />

− bc −c a<br />

2 3<br />

(c) Find <strong>the</strong> inverse of and check your answer.<br />

1 4<br />

(6) Suppose you were on a desert island and needed to invert a matrix. Here’s<br />

a procedure you could use. You could also use this procedure in your own<br />

computer program if you needed to invert a matrix.<br />

(a) Construct a 5 × 5 matrix A of rank 5 in Octave. Check that <strong>the</strong><br />

row-reduced form of A is I.<br />

(b) Form <strong>the</strong> double-wide matrix B = [A| I] with <strong>the</strong> command B=[A,eye(5)].<br />

(c) Row-reduce B to a matrix C: C=rref(B).<br />

Date: March 18, 2010.<br />

1


2 DAVID MEREDITH<br />

(d) You should get C = [I| D]. The matrix D is A −1 . Recover D with<br />

<strong>the</strong> command: D = C(:,6:10) and check that AD = DA = I.<br />

Department of Ma<strong>the</strong>matics<br />

San Francisco State University<br />

San Francisco, CA 94<strong>13</strong>2<br />

E-mail address: meredith@sfsu.edu<br />

URL: http://online.sfsu.edu/~meredith

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