Multiplying Matrices

Scalar Multiplication - each element in a matrix is multiplied by a constant.

**Multiply rows times columns. **You can only multiply if the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix. They must match. Dimensions: 3 x 2 2 x 3 The dimensions of your answer.

Examples: 2(3) + -1(5) 2(-9) + -1(7) 2(2) + -1(-6) 3(3) + 4(5) 3(-9) + 4(7) 3(2) + 4(-6)

*They don’t match so can’t be multiplied together.* Dimensions: 2 x 3 2 x 2 *They don’t match so can’t be multiplied together.*

2 x 2 2 x 2 *Answer should be a 2 x 2 0(4) + (-1)(-2) 0(-3) + (-1)(5) 1(4) + 0(-2) 1(-3) +0(5)

Determinants

**To find a determinant you must have a SQUARE MATRIX!!** Determinant - a square array of numbers or variables enclosed between parallel vertical bars. **To find a determinant you must have a SQUARE MATRIX!!** Finding a 2 x 2 determinant:

Find the determinant:

Finding a 3x3 determinant: Diagonal method Step 1: Rewrite first two columns of the matrix.

126 - (-52) 126 + 52 = 178 Step 2: multiply diagonals going up! -224 +10 +162 = -52 Step 2: multiply diagonals going up! Step 3: multiply diagonals going down! +12 -126 +240 =126 Step 4: Bottom minus top! 126 - (-52) 126 + 52 = 178

38 - 38 = 0 Step 1: multiply diagonals going up! -18 +50 +6 = 38 Step 1: multiply diagonals going up! Step 2: multiply diagonals going down! - 15 45 + 8 = 38 Step 3: Bottom minus top! 38 - 38 = 0

Expansion of a Third-Order Determinant

example: Evaluate using expansion by minors.

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