1. 14.1 Matrix Addition and Scalar Multiplication OBJ:  To find the sum, difference, or scalar multiples of matrices

2. EX:  An automobile dealer sells four different models whose fuel economy is shown in the table below: This information can be displayed as a rectangular array of numbers enclosed by brackets, called a matrix (plural, matrices), usually labeled with a capital letter. Spts Car Se- dan Sta- tion Wag Van City Mpg 17 221716 High- way Mpg 233024 19

3. sp se sw v M =  17 22 17 16  c  23 30 24 19  h Each number is an element (or entry) of the matrix. The dimensions are the number of rows and columns. Since M has two rows and four columns, M is a 2 x 4 matrix, denoted by M 2x4. It is a “driving-condition by model” matrix.

4. If the rows and columns are interchanged, you get the transpose of M, denoted by M t c h M t =   sp l l se l l sw   v M t 4x2 is a “model by driving-condition” matrix, with 4 rows and 2 columns.

5. If the rows and columns are interchanged, you get the transpose of M, denoted by M t c h M t =  17 23  sp l 22 30 l se l 17 24 l sw  16 19  v M t 4x2 is a “model by driving-condition” matrix, with 4 rows and 2 columns.

6. The Environmental Protection Agency mandated in 5 years the fuel performance figures must increase 10%. This means every element in matrix M must be multiplied by 1.10, resulting in the matrix sp se sw v 1.1M =  18. 7 24.2 18.7 17.6  c  25.3 33 26.5 20.9  h This is called scalar multiplication, with 1.1 being called a scalar.

7. EX:  If A =  3 1 5 , find At, 2A, and -3A  4 0 -2  At=At=  3 4  | 1 0   5 -2  2A =  6 2 10   8 0 -4  -3A =  -9 -3 -15   -12 0 6 

8. Two matrices with the same dimensions can be added or subtracted, by finding the sums or differences of the corresponding elements. EX:  A =  381   40 -3   -215  B =  209   4 -6 -5   0 72  Find A + B and A – B. A + B =  5 8 10   8 -6 -8   -2 8 7  A – B =  1 8 -8   0 6 2   -2 -6 3 

9. EX:  A =  2 -1   4 0   0 -8  B =  -6 3 5   0 7 -4  Find A t + B and A + B t. A t =  2 4 0   -1 0 -8  B t =  -6 0   3 7   5 -4  A t + B =  -4 7 5   -1 7 -12  A + B t =  -4 -1   7 7   5 -12 

10. Two matrices are equal if and only if they have the same dimensions and all corresponding elements (same row, same column) are equal. EX:  Find the values of the variables for which the given statement is true.  a b  –  2 -3  =  7 2.5   c d   5 -1   -1 0   a b  =  7 2.5  +  2 -3   c d   -1 0   5 -1   a b  =  9 -.5   c d   4 -1 

11. Solve the matrix equation for X 2  5 1  + 3X =  1 -4   3 4   3 -7   10 2  + 3X =  1 -4   6 8   3 -7  3X=  1 -4  –  10 2  =  3 -7   6 8  _1_  -9 -6  3  -3 -15  X =  -3 -2   -1 -5 