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Fundamentals of Engineering Analysis EGR 1302 - Determinants. 3 positive terms. 3 negative terms. Determinants. - A Property of a Square Matrix. “Eyeball” Method. Determinant of a 3x3. Let’s factor out the elements of the first row of the matrix, i.e.

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## Fundamentals of Engineering Analysis EGR 1302 - Determinants

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**Fundamentals of Engineering Analysis**EGR 1302 - Determinants**3 positive terms**3 negative terms Determinants - A Property of a Square Matrix “Eyeball” Method**Determinant of a 3x3**Let’s factor out the elements of the first row of the matrix, i.e.**We can identify this construct as the “Cofactor”**Determinant of a 3x3**Every element in a square matrix has a cofactor**The Cofactor Matrix of a 3x3 The cofactor of any element is “the determinant formed by striking out the Row & Column of that element**Sign of the Cofactor:**The Cofactor Matrix of a 3x3 Caution: Do not forget the signs of the cofactors**Determinant by Row Expansion**using the first row: Row Expansion:**Using the TI-89 to find Determinants**We had previously entered a matrix and assigned it to the variable “a” The calculator has the built-in function “det()“ Which calculates the determinant of a square matrix.**Determinant by Row or Column Expansion**Select Any Row or Column to do the Expansion Pick Column #1 to simplify the calculation due to the zero terms.**Finding the Cofactor Matrix of A**Calculators and Computers obviously make this process easier. **Similar, but not quite**Rules for 2x2 Inverse and the Cofactor Matrix 1. Swap Main Diagonal 2. Change Signs on a12, a21 3. Divide by detA**Properties of Determinants**1. Determinant of the Transpose Matrix det A = det AT**for**Properties of Determinants 2. Multiply a single Row (Column) by a Scalar - k det B = k*det A det B = 3*det A**swap**4. Expansion by any Rows (Columns) equals the same Determinant Properties of Determinants 3. If two Rows (Columns) are swapped, the sign changes det B = -det A Recall:**det B = 0**det A = 0 Col2 = 2*Col1 Row2 = Row1 Properties of Determinants 5. If two Rows (Columns) are equal, or the same ratio, i.e., Row1 = k*Row2 det A = 0 The matrix A is “singular” But if detA=0, a unique solution does not exist Recall Rule #3 to find A-1, divide by detA**Construct D by creating a new Row 2**Properties of Determinants • If a new matrix B is constructed from A • by adding K*rowj to another rowi … det B = det A These are called Row (Column) Operations**Finding the Determinant: Two Methods**“Eyeball” Method 2 + (-40) + (-6) – (-5) -12 –(-8) = -43 Row Expansion 1*(2-12) -2(-4+3) -5(8-1) -10 + 2 -35 = -43

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