Fundamentals of Engineering Analysis Matrices and Determinants Part 1 Eng. Hassan S. Migdadi

Objectives: Evaluate the determinant of a 3 x 3 matrix, and Find the area of a triangle given the coordinates of its vertices.

Determinants Every square matrix has a determinant. The determinant has the same elements as the matrix, but they are enclosed between vertical bars instead of brackets. you have learned a method for evaluating a 2 x 2 determinant.

Determinants The determinant of is . To evaluate The determinant, use the rule for second order determinants.

Expansion by Minors A method called expansion by minors can be used to evaluate the determinant of a 3 x 3 matrix. The minor of an element is the determinant formed when the row and column containing that element are deleted. For the determinant

Expansion of a Third-Order Determinant Expansion by Minors To use expansion by minors with third-order determinants, each member of one row is multiplied by its minor. The signs of the products alternate, beginning with the second product. The definition below shows an expansion using the elements in the first for of the determinant. However, ANY row can be used. Expansion of a Third-Order Determinant

Ex. 1: Evaluate the determinant of Using expansion by minors.

Using Diagonals Another method for evaluating a third order determinant is using diagonals. STEP 1: You begin by repeating the first two columns on the right side of the determinant.

aei bfg cdh Using Diagonals STEP 2: Draw a diagonal from each element in the top row diagonally downward. Find the product of the numbers on each diagonal. aei bfg cdh

idb gec hfa Using Diagonals STEP 3: Then draw a diagonal from each element in the bottom row diagonally upward. Find the product of the numbers on each . idb gec hfa

Using Diagonals To find the value of the determinant, add the products in the first set of diagonals, and then subtract the products from the second set of diagonals. The value is: aei + bfg + cdh – gec – hfa – idb

First, rewrite the first two columns along side the determinant. Ex. 2: Evaluate using diagonals. First, rewrite the first two columns along side the determinant.

Next, find the values using the diagonals. Ex. 2: Evaluate using diagonals. Next, find the values using the diagonals. -5 24 4 60 Now add the bottom products and subtract the top products. 4 + 60 + 0 – 0 – (-5) – 24 = 45. The value of the determinant is 45.

and then finding |A|, since the area cannot be negative. Area of a triangle Determinants can be used to find the area of a triangle when you know the coordinates of the three vertices. The area of a triangle whose vertices have coordinates (a, b), (c, d), (e, f) can be found by using the formula: and then finding |A|, since the area cannot be negative.

Ex. 3: Find the area of the triangle whose vertices have coordinates (-4, -1), (3, 2), (4, 6). How to start: Assign values to a, b, c, d, e, and f and substitute them into the area formula and evaluate. a = -4, b = -1, c = 3, d = 2, e = 4, f = 6 -3 8 -24 -8 -4 18 Now add the bottom products and subtract the top products. -8 + (-4) + 18 – 8 – (-24) –(-3) = 25. The value of the determinant is 25. Applied to the area formula ½ (25) = 12.5. The area of the triangle is 12.5 square units.

Sometimes one or more of the elements of a determinant may be unknown, but the value of the determinant is known. You can use expansion to find the values of the variable. Ex. 4: Solve for n if 24n – 84 – 36n + 48n +84n – 18 = -582 120n – 102 = -582 120n = -480 n = -4 The value of n is -4. Collect like terms Add 102 to both sides Divide by 120 both sides.