By Steven A. Leduc

ISBN-10: 0822053314

ISBN-13: 9780822053316

A complete assessment consultant that can assist you refresh your examine. This advisor is very precious for midterms and ultimate assessments, condensing a semester's worthy of knowledge into one concise quantity.

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**Example text**

If not, give th e equation of the plane which is parallel to P that does contai n the origin . The vectors pq = (0, 3, -3) and pr = '(2, 2, -1) lie in P , and their cross product, pq x pr, is normal to P. See Figure 34 . 2—3•(—2)) _ (3, 6, 6 ) LINEAR ALGEBRA 45 VECTOR ALGEBRA Since the vector v = (3, 6, 6) is normal to P, the standar d equation of P is given b y 3x+6y+6z= d for some constant d . Substituting the coordinates of any of the three given points (p, q, or r) into this equation yields d = 12 .

Since only matrices of the same size can be added, onl y the sum F + H is defined (G cannot be added to either F o r H) . The sum of F and H i s [1 1 6 2+1 -1+6 3 5 F+H= 3 0 + -1 -2 = 3—1 0—2 = 2 - 2 -5 2 0 -3 -5+0 2—3 -5 - 1 2 -1 _ n Since addition of real numbers is commutative, it follows that addition of matrices (when it is defined) is also commutative; that is, for any matrices A and B of the same size, A + B will always equal B + A . 54 CLIFFS QUICK REVIEW MATRIX ALGEBR A Example 7 : If any matrix A is added to the zero matrix of th e same size, the result is clearly equal to A : A +0=0+A = A This is the matrix analog of the statement a + 0 = 0 + a = a, which expresses the fact that the number 0 is the additive iden■ tity in the set of real numbers .

Since every matrix in MZX3(R ) consists of 2 rows and 3 columns, A will contain 2 x 3 = 6 entries . An example of suc h a matrix is A 10 3 -2 4 1 ■ Example 2 : If B is the 2 x 2 matrix whose (i, j) entry is given by the formula bu = (—1)'+'(i +j), explicitly determine B. The (1, 1) entry of B is b ll = (—1)' +'(1 + 1) = 2 ; the (1, 2 ) entry is b l2 = (—1) 1+2(1 + 2) = -3 ; the (2, 1) entry is also -3 ; and the (2, 2) entry is b 22 = (—1)Z+2 (2 + 2) = 4 . ) The (1, 1), (2, 2), and (3, 3) entries are each equal to 1, but al l other entries are 0 .

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