Vocab: Matrices – Determinants and Inverses (Intro) Terms: Identity matrix – square matrix with ‘1’s in the main diagonal and zeros in every other entry. Examples: zero matrix – matrix in which all entries are ‘0’. Examples a.b. a. b.

Vocab: Matrices – Determinants and Inverses (Intro) Terms: Main diagonal – starting from element in upper left (a 1,1 ) and going down to right (a n,n ), where ‘n’ = nbr of rows/columns. Other ‘main’ diagonals starting a 1,2 and going down to right. Off diagonal – starting from element in lower left (a n,1 ) and going up to right (a 1,n ), where ‘n’ = nbr of rows/columns Determinant of a matrix – the sum of the products of the elements in the main diagonals, minus the products of the elements in the OFF diagonals.

Vocab: Matrices – Determinants and Inverses (Intro) Terms: Transpose of a matrix – switching the columns with the rows and all of the associated entries. Ex: Given A= |A| represents determinant of A Notation of a determinant – putting bars, similar to the absolute bars, around the values of a matrix.

EXAMPLE 1 Transposing Matrices - examples (switch rows and columns). a SOLUTION c – b – – – – a b. c –

Steps: Evaluate determinants 1.For any matrix larger than a 2 x 2, repeat the first ‘n-1’ columns, where ‘n’ = the number of columns, to the right of matrix. This creates an augmented matrix. 2.Identify the main diagonals and off-main diagonals. 3.Calculate the product of each of the main, and off-main diagonals. 4.Sum the products of the main diagonals, and subtract the sum of the products of the off-main diagonals.

EXAMPLE 1 Evaluate determinants Evaluate the determinant of the matrix. a SOLUTION b – – – –

GUIDED PRACTICE for Examples 1 and – = 3(1) 6 ( 2) – – = = – – – – Off-mains: – = 21 – Evaluate the determinant of the matrix – – -3 –8 + Mains: = 22

GUIDED PRACTICE for Examples 1 and – – – – – – – – – – – = ( ( 42) (0 + ( 280) + 8) – – – = 470

Notes Finding area of a triangle, given the three vertices. Steps: 1.Create a 3x3 matrix, call it matrix B. 2.Place the x-coordinates of each vertices in column ‘1’ of the matrix. 3.Place each corresponding y-coordinate in the second column next to its associated x-coord. 4.Place all ‘1’s in the third column. 5.Use the following formula: 6.NOTE: the absolute bars on the outside, force the answer to be positive, as expected for ‘area’. Area = x1x1 x3x3 x2x2 y1y1 y2y y3y3 1 2 x1x1 x3x3 x2x2 y1y1 y2y2 y3y3

GUIDED PRACTICE Finding area of a triangle 4. Find the area of the triangle with vertices A(5, 11), B(9, 2), and C(1, 3). The coordinate of the vertices of the triangular region are A(5, 11), B(9, 2), and C(1, 3).so the area of region is. Area = – – 1 2 = [( ) ( )] – = 34= 68 2

GUIDED PRACTICE for Examples 1 and 2 The area of the region is about 34 ANSWER

EXAMPLE 2 Find the area of a triangular region Sea Lions Off the coast of California lies a triangular region of the Pacific Ocean where huge populations of sea lions and seals live. The triangle is formed by imaginary lines connecting Bodega Bay, the Farallon Islands, and Año Nuevo Island, as shown. (In the map, the coordinates are measured in miles.) Use a determinant to estimate the area of the region.

EXAMPLE 2 Find the area of a triangular region SOLUTION The approximate coordinates of the vertices of the triangular region are ( 1, 41), (38, 43), and (0, 0). So, the area of the region is: – – Area = – – 0 + – 1 2 – = [( ) ( )] – = The area of the region is about 758 square miles – – – – 0 38 – =

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