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

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 High- way Mpg

sp se sw v M =   c   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.

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.

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.

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 =   c   h This is called scalar multiplication, with 1.1 being called a scalar.

EX:  If A =  , find At, 2A, and -3A   At=At=  3 4  | 1 0   5 -2  2A =     -3A =    

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

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

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  =    c d   5 -1   -1 0   a b  =   +  2 -3   c d   -1 0   5 -1   a b  =    c d   4 -1 

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_   3   X =    