MATRIX OPERATIONS Summary of article: What is an operation? Addition of two matrices. Multiplication of a Matrix by a scalar. Subtraction of two matrices: two ways to do it. Combinations of Addition, Subtraction, Scalar Multiplication. Matrix Multiplication An operation is a way that we combine two elements. The basic operations are: Addition (+) Subtraction (-) Multiplication ( * ) Division ( ÷ ) Here are some examples of combining elements and the operations used: Combining elements that are rational numbers (fractions) using the operation of addition . 1 2 + 3 4 Combining elements that are second degree binomials using the operation of multiplication. (2x2+ 3)(4 x 2 − 1) Combining elements that are integers using the operation of division . −35 ÷ 7 This FAQ will review the definitions of addition, scalar multiplication, subtraction, and the multiplication of matrices. Division is undefined for matrices, but there is a separate related concept, Inverse Matrices, that is similar to division and is found among the other FAQ topics listed on the Online Math Center. ADDITION of TWO MATRICES To add two matrices, their orders (the number of rows and columns in both matrices ) must be the same. Add the corresponding row-column elements from each Matrix to produce a new element in the same row-column location. Example 1: Both matrices are order 2 x 2 31 2 8 3 + 2 18+ 5 9 A= B = AB += = 4−6− 3 − 1 4 + − 3 −+− 61 1 −7 Example 2: Both matrices are order 3 x 5 52071 1 46− 2 − 1 A=−310912 B =−−− 4 20 1 9 1 8−210−3 − 2 − 29− 7 5 5+ 1 24+7+ − 2 06 + 1+ − 1 6 66 5 0 +=−+− + +−+ +−=− A B 34120 1012 9 917 21 9 11 8 8+ − 2−+−2 20+ − 7 1 +9−3 + 5 6 − 4− 7 1 0 2 MULTIPLICATION of a MATRIX by a scalar A scalar is simply a number. To multiply a Matrix by a scalar, distribute the scalar to all elements in the Matrix and multiply. Example 1: Multiply Matrix A by the scalar 3: 421− 421 −3 *4*2* 3 − 3 1 12 − 63 A=810 3 A ==3 810 333 *8 *1 * 0 = 243 0 297 2973 *2 3 *9 3 *7 62721 1 Example 2: Multiply Matrix B by the scalar : 4 1 1 *4 *(− 20) 4− 20 1 4 4 1− 5 B = B = = 16 8 4 1 1 4 2 *16 *8 4 4 SUBTRACTION of TWO MATRICES To subtract two matrices, their orders (the number of rows and columns in both matrices ) must be the same. The easy way… To subtract Matrix A and B, simply subtract corresponding row-column elements. Don’t forget to change signs of elements in Matrix B Example 1: Subtract Matrix B from Matrix A. Both are order 2 x 2. 31 2 8 3 − 2 18− 1 −7 A= B = AB −= = 4−6 − 3− 1 4 − − 3− 61 −−− 7 5 Example 2: Subtract Matrix B from Matrix A. Both are order 1 x 4. A=[2 −4710] B = [− 1 1 − 3 9 ] A−=− B [][]24710 − − 1 1 − 3 9 =−[]2− 1 −−4 1 7 − − 3 10− 9 = []3 −5 10 1 Example 3: Subtract D from C 167− − 2 C=1225 D = 3 9− 35 15 Matrix C is a 3 x 3 order Matrix. Matrix D is a 3 x 1 order Matrix. Since the order of the two matrices is different, they can NOT be subtracted. Now the mathematics behind the scenes… The negative sign in front of the second Matrix is actually the scalar “ −1”. Distribute the scalar “ −1” to all elements in the second Matrix. Add the corresponding row-column elements from each Matrix to produce a new element in the same row-column location. Example: Subtract B from A 83 76− = − =− A25 B 12 −91 30 AB− = A +− 1* B 83 76− 83 −1 *7 − 1 *(6) − = −+− − = −+− − − 251 *12 25 1 *(1) 1 *2 −91 30 − 91−1 *3 − 1 *0 83 − 76 8736 +− + 1 9 =25 −+ 12 −= 2 + 1−+− 52 = 3 − 7 −9 1 − 3 0 −+−9 3 10 + − 121 COMBINATIONS of ADDITION, SUBTRACTION, and Scalar MULTIPLICATION. The order of operations requires multiplication be done before addition or subtraction, so first multiply the elements inside a Matrix by the scalar in front of it. Add and/or subtract afterwards. Example: 34− 17 A= B = 25 83− Find 3A− 5 B 3− 4 3*33*4 − 9 − 12 3A = 3 = = 2 5 3*2 3*5 6 15 1 7 5*1 5*7 5 35 5B = 5 = = 8− 3 5*8 5*3 − 40 − 15 9− 12 5 35 3A− 5 B = − 6 15 40− 15 95−−− 1235 4 − 47 = = 6− 40 15 −− 15 − 34 30 MULTIPLICATION of TWO MATRICES Two matrices Aand B can be multiplied if the number of columns in A is the same as the number of rows in B. The new Matrix will have the same number of rows as A and the same number of columns as B. Example, find A* B 7− 3 14 2 5 9 A= B = 5 1 9 1− 6 5 −4 2 3 Matrix A is 2 x 3 (2 rows, 3 columns ). Matrix B is a 3 x 3 ( 3 rows , 3 columns) Matrix. The columns in A equal the rows in B, so we can multiply A*B producing a 2 x 3 Matrix. However, the columns in B do not equal the rows in A, so we can not multiply B*A. Steps in multiplying two matrices 1. Determine if the two matrices can be multiplied, i.e. the number of columns in A equals the number of rows in B. If the order of A is (m x n) and the order of B is (n x p), the new Matrix will be of order (m x p). 2. Set up the new, blank (m x p) Matrix. 3. Pick a row-column location of an element in the new Matrix, e.g. the element in row 1 column 1 of the new Matrix. 4. Multiply the first element from the identified row in A by the first element in the identified column of B. Multiply the second element from the identified row of A by the second element in the identified column of B. Continue across the row of A and down the column of B. Then add all of the results. Place the answer in the new Matrix at the row-column location identified. 5. Continue until all row-column locations of the new Matrix are filled. 34− 17 Example: A= B = 25 83− Matrix A has 2 rows and 2 columns; Matrix B has 2 rows and 2 columns. The number of columns in A equals the number of rows in B, so the two matrices can be multiplied. The (2 x 2) times ( 2 x 2) will produce a new (2 x 2) Matrix. Set up the new, blank 2 x 2 Matrix. _ _ ? A* B = = _ _ The question mark has been placed in the first row, first column location of the new Matrix. So multiply the first element in row 1 of Matrix A by the first element in column 1 of Matrix B. Then multiply the second element in row 1 of Matrix A by the second element in column 1 of Matrix B. 3− 4 1 7 A= B = 2 5 8 − 3 Multplying row 1 of A by column 1 of B: (31 )( )+ (− 48 )( ) We have gone across row 1 in Matrix A and down column 1 in Matrix B, so we can add the results and place the answer in row 1-column 1 of the new Matrix. −29 _ (31 )( )+ ( − 48)( ) =3 − 32 =− 29 New Matrix: row 1, column 1: _ _ −29 ? Now let’s find the element for row 1, column 2: _ _ 3− 4 1 7 A= B = 2 5 8 −3 Multplying row 1 of A by column 2 of B: (3 )(7 ) + (−4 )(− 3 )= 21 + 12 = 33 So row 1, column 2 of the new Matrix is the element 33: −29 33 ? ? For row 2, column 1 multiply row 2 of Matrix A and column 1 of Matrix B. For row 2, column 2 multiply row 2 of Matrix A and column 2 of matrix B: 3− 4 1 7 A= B = (2)(1)+ (5) (8) =+=2 40 42 25 8 −3 3− 4 1 7 A= B = (2)( 7) +(5)( − 3) =−=−14 1 5 1 25 8 − 3 Hence: −29 33 A* B = 42 −1 4 Example : Find A * B A=[]153 B = − 2 1 A is order (1 x 3) and B is order (3 x 1), so the new matrix will be order (1 x 1): [?] There is only one element in the new matrix: the row 1, column 1 element . Multiply row 1 elements of Matrix A by column 1 elements of Matrix B and add the result. 4 A=[]1 5 3 B =− 2 (1)(4)+ (5)( − 2) + (3)(1) = 4 −10 + 3 =− 3 1 A * B = [−3] 5 0 1 0 4 Example : Find A * B A= B = 1 6 −1 2 3 7− 2 A is order (2 x 3) and B is order (3 x 2), so the new matrix will be order (2 x 2): row1, column 1 row 1, column 2 row2, column 1 row 2, column 2 Row 1, Column 1: (1)(5)+ (0)(1) + (4)(7) =++= 5 0 28 33 Row 1, Column 2: (1)(0)+ (0)(6) + (4)( −=+−=− 2) 0 0 8 8 Row 2, Column 1: (− 1)(5) + (2)(1) + (3)(7) =−++= 5 2 21 18 Row 2, Column 2: (− 1)(0) + (2)(6) + (3)( −=+−= 2) 0 12 6 6 33− 8 A * B = 18 6 .