2 2 Matrices. Scalar multiplication for matrices. A2 2matrix(pronounced 2-by-2matrix )isasquareblockof4numbers. For example,

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1 2 2 Matrices A2 2matrix(prononced 2-by-2matrix )isasqareblockofnmbers. For example, is a 2 2matrix. It scalleda2 2 matrix becase it has 2 rows 2 colmns. The for nmbers in a 2 2matrixarecalledtheentries of the matrix. Two matrices are eqal if the entry in any position of the one matrix eqals the entry in the same position of the other matrix. Examples Scalar mltiplication for matrices To take the prodct of a scalar a matrix, jst as with vectors, mltiply every nmber in the matrix by the scalar. For example, * * * * * * * * * * * * * 270

2 Mltiplying a matrix a vector Sppose A is a 2 2matrix.Tobemoreprecise,let ssaythat a b A c d where a, b, c, d 2 R. The matrix A a vector 2 R 2 can be combined to prodce a new vector in R 2 as follows: Imagine the matrix A as two rows of nmbers a b (a, b) 7! c d (c, d) To find the prodct a b c d w we simply mltiply the first row, (a, b), the colmn to obtain the nmber (a, b) a + bw. That will be the first entry for the new vector in R 2. The second entry in the new vector will be (c, d) c + dw, orin other words, the second entry in or new vector inr 2 will be the prodct of the second row of the matrix A with the colmn. Or new vector in R 2 will then be the colmn a + bw c + dw To smmarize, we have a b c d w Example a + bw c + dw

3 Matrices as fnctions Notice that if is a 2 2matrix,if a b c d 2 R 2 is a vector, then their prodct a b c d w a + bw c + dw is also a vector in R 2. Therefore, if we fix a particlar 2 2matrix,itdefinesawaytoassignto any vector in R 2 anewvectorinr 2. That is, any 2 2matrixdescribesa fnction whose domain is R 2 whose target is R 2. Example. Let B 5 3 Then B can be thoght of as a fnction B : R 2! R 2. The vector 6 is in the domain of the matrix fnction B. If we pt this vector into B, we will get ot the vector 16 2 since, as we have seen above, Identity matrix Notice that x y 1 x +0 y x 0 x +1 y y 272

4 That means that any vector we pt into the matrix fnction gets retrned to s naltered. This is the identity fnction whose domain is the set R 2, so we call the 2 2 identity matrix. * * * * * * * * * * * * * Matrix mltiplication Yo can mltiply two 2 2matricestoobtainanother2 2matrix. Order the colmns of a matrix from left to right, so that the 1 st colmn is on the left the 2 nd colmn is on the right. To mltiply two matrices, call the colmns of the matrix on the right inpt colmns, pt each of the inpt colmns into the matrix on the left (thinking of it as a fnction). The colmn that is assigned to the 1 st inpt colmn by the matrix fnction will be the 1 st colmn of the prodct yo are trying to find. The colmn that is assigned to the 2 nd inpt colmn by the matrix fnction will be the 2 nd colmn of the prodct. Let s try an example. To find the 2 2matrixthateqalstheprodct first divide the matrix on the right into colmns ! Enter the leftmost colmn into the matrix fnction

5 the reslt is This will be the first colmn of the prodct And it s second colmn will eqal Pt the two colmns together to find that Matrix mltiplication is fnction composition Let A B be 2 2matrices.WehaveseenthatthematrixA defines a fnction A : R 2! R 2, that there is also a fnction B : R 2! R 2. Thinking of A B as fnctions, ignoring that they are matrices, we cold compose them to obtain a new fnction A B. This new fnction is also described by a matrix the matrix AB, where AB means the matrix mltiplication of A with B. Becase fnction composition isn t commtative, neither is matrix mltiplication. Try this yorself: write down two 2 2matrices,A B. Probably AB 6 BA. * * * * * * * * * * * * * Inverse matrices Again let A be a 2 2matrix.ThenthereisafnctionA : R 2! R 2. Thinking of A prely as a fnction, not as a matrix, the inverse of A is a fnction A 1 : R 2! R 2 sch that both A A 1 A 1 A are the identity fnction. 27

6 To translate the above paragraph from fnctions to matrices, let s now think of A as a matrix again. Remember that fnction composition is really matrix mltiplication, that the matrix that represents the identity fnction is the identity matrix. After making these translations, we are left with the definition of the inverse of a 2 2matrixA as another 2 2matrixA 1 sch that AA 0 A 1 A Example. We can see that 1 by checking that * * * * * * * * * * * * * 275

7 Exercises 1.) The matrix A describes a fnction A : R 2! R 2. Find the vectors ) The matrix B describes a fnction B : R 2! R 2. Find the vectors Find the following prodcts of matrices: 3.) ) ) 3 2 For #6 #7, determine whether the two matrices given are inverses of each other. 6.) )

8 For #8-11, match the rational fnctions with their graphs. 8.) 2(x 2)(x )(x 7) (x+3) 3Z 9.) 2(x+2)(x+) 2 (x+7) (x+3) 10.) 2(x+2)(x )(x 7) (x 3) 1.) 2(x+2)(x )(x+7) (x+3) L77 A.) B.) -7 3Z L77 C.) D.) -7 3Z