§ 3.5 Determinants and Cramer’s Rule

Determinants Evaluate the determinant of the matrix. EXAMPLE SOLUTION Blitzer, Intermediate Algebra, 5e – Slide #2 Section 3.5

Determinants The determinant of the matrix is defined by Definition of the Determinant of a 2 by 2 matrix The determinant of the matrix is defined by The determinant of a matrix may be positive or negative. The determinant can also have 0 as its value. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 3.5

Solving a Linear System in Two Variables Using Determinants Determinants – Cramer’s Rule Solving a Linear System in Two Variables Using Determinants Cramer’s Rule If then and where Blitzer, Intermediate Algebra, 5e – Slide #4 Section 3.5

Determinants – Cramer’s Rule EXAMPLE Use Cramer’s rule to solve the system: SOLUTION Because we’re using Cramer’s rule to solve this system, we must first write the system in standard form. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 3.5

Determinants – Cramer’s Rule CONTINUED Now we need to determine the different values as defined in Cramer’s rule: Therefore Blitzer, Intermediate Algebra, 5e – Slide #6 Section 3.5

Determinants – Cramer’s Rule CONTINUED Now we can determine x and y. Therefore, the solution to the system is (-2,-1). As always, the ordered pair should be checked by substituting these values into the original equations. Blitzer, Intermediate Algebra, 5e – Slide #7 Section 3.5

Evaluating the Determinant of a 3x3 Matrix Determinants Evaluating the Determinant of a 3x3 Matrix 1) Each of the three terms in the definition contains two factors – a numerical factor and a second-order determinant. 2) The numerical factor in each term is an element from the first column of the third-order determinant. 3) The minus sign precedes the second term. 4) The second-order determinant that appears in each term is obtained by crossing out the row and the column containing the numerical factor. The minor of an element is the determinant that remains after deleting the row and column of that element. For this reason, we call this method expansion by minors. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 3.5

Determinants Evaluate the determinant of the matrix. EXAMPLE Evaluate the determinant of the matrix. SOLUTION The minor for 2 is . The minor for -2 is . The minor for 4 is . Blitzer, Intermediate Algebra, 5e – Slide #9 Section 3.5

Determinants Therefore CONTINUED Therefore = 2[(3)(8) – (-6)(5)] + 2[(1)(8) – (-6)(0)] + 4[(1)(5) – (3)(0)] = 2[24 – (-30)] + 2[8 – 0] + 4[5 – 0] = 2[24 + 30] + 2[8 – 0] + 4[5 – 0] = 2[54] + 2[8] + 4[5] = 108 + 16 + 20 = 144 Therefore, the determinant of the matrix is 144. Blitzer, Intermediate Algebra, 5e – Slide #10 Section 3.5

Determinants – Cramer’s Rule EXAMPLE Use Cramer’s rule to solve the system: SOLUTION We first need to determine the different values as defined in Cramer’s rule: Blitzer, Intermediate Algebra, 5e – Slide #11 Section 3.5

Determinants – Cramer’s Rule CONTINUED Therefore Blitzer, Intermediate Algebra, 5e – Slide #12 Section 3.5

Determinants – Cramer’s Rule CONTINUED Now I can determine x, y and z. Therefore, the solution to the system is (-2, 3/5, 12/5). As always, the ordered pair should be checked by substituting these values into the original equations. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 3.5