College Algebra Chapter 6 Matrices and Determinants and Applications Section 6.3 Operations on Matrices

Concepts 1. Determine the Order of a Matrix 2. Add and Subtract Matrices 3. Multiply a Matrix by a Scalar 4. Multiply Matrices 5. Apply Operations on Matrices

Determine the Order of a Matrix The order of a matrix is determined by the number of rows and number of columns. A matrix with m rows and n columns is an m  n matrix.

Examples 1 – 4: Determine the order of the matrix.

Determine the Order of a Matrix A matrix can be represented generically by The notation aij represents the element in the ith row, jth column. For example, a23 represents the element in the 2nd row, 3rd column.

Examples 5 – 7: Determine the value of the given element of the matrix 5. a12 = _____ 6. a22 = _____ 7. a14 = _____

Concepts 1. Determine the Order of a Matrix 2. Add and Subtract Matrices 3. Multiply a Matrix by a Scalar 4. Multiply Matrices 5. Apply Operations on Matrices

Add and Subtract Matrices To add or subtract two matrices, the matrices must have the same order, and the sum or difference is found by adding or subtracting the corresponding elements.

Example 8:

Example 9:

Add and Subtract Matrices The additive inverse of a matrix is found by taking the opposite of each element in the matrix.

Example 10: Given , find the additive inverse, –A.

Add and Subtract Matrices A matrix in which all elements are zero is called a zero matrix and is denoted by O. The sum of a matrix A and its additive inverse –A is the zero matrix of the same order.

Add and Subtract Matrices Two matrices are equal if and only if they have the same dimensions and if their corresponding elements are equal.

Add and Subtract Matrices Properties of Matrix Addition Let A, B, and C be matrices of order m  n, and let O be the zero matrix of order m  n. Then, 1. A + B = B + A Commutative property 2. A + (B + C) = (A + B) + C Associative property 3. A + (–A) = O Inverse property 4. A + O = O + A = A Identity property

Concepts 1. Determine the Order of a Matrix 2. Add and Subtract Matrices 3. Multiply a Matrix by a Scalar 4. Multiply Matrices 5. Apply Operations on Matrices

Multiply a Matrix by a Scalar The product of a real number k and a matrix is called scalar multiplication. The real number k is called a scalar to distinguish it from a matrix.   To multiply a matrix A by a scalar k, multiply each element in the matrix by k.

Example 11:

Example 12:

Multiply a Matrix by a Scalar Properties of scalar multiplication Let A and B be m  n matrices and let c and d be real numbers. Then, 1. c(A + B) = cA + c B Distributive property of scalar multiplication 2. (c + d)A = cA + dA Distributive property 3. c(dA) = (cd)A Associative property

Example 13: Given the matrices A and B, solve for X.

Concepts 1. Determine the Order of a Matrix 2. Add and Subtract Matrices 3. Multiply a Matrix by a Scalar 4. Multiply Matrices 5. Apply Operations on Matrices

Multiply Matrices Given matrices A and B, to multiply AB we require that the number of columns in A be equal to the number of rows in B. The resulting matrix will have dimensions equal to the number of rows of A by the number of columns of B. If , we can multiply AB, we cannot multiply BA.

Example 14:

Multiply Matrices Let A be an m  p matrix and B be a p  n matrix, then the product AB is an m  n matrix.   For the matrix AB, the element in the ith row and jth column is the sum of the products of the corresponding elements in the ith row of A and the jth column of B. If the number of columns in A does not equal the number of rows in B, then the product AB is not defined.

Multiply Matrices

Example 15:

Concepts 1. Determine the Order of a Matrix 2. Add and Subtract Matrices 3. Multiply a Matrix by a Scalar 4. Multiply Matrices 5. Apply Operations on Matrices

The price of each burger combination is given by the matrix B. Example 16: Bianca's Burger Basket sells three kinds of burgers with three kinds of buns. The number of burger combinations sold is represented by matrix A. wheat rye white Turkey burger Veggie burger Beef burger The price of each burger combination is given by the matrix B. turkey veggie beef

Find the product BA and interpret its meaning. Example 16 continued: Find the product BA and interpret its meaning. http://www.kingarthurflour.com/recipes/beautiful-burger-buns-recipe