M1410804 - (Section 8.4: The Determinant of a Square...

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(Section 8.4: The Determinant of a Square Matrix) 8.57 SECTION 8.4: THE DETERMINANT OF A SQUARE MATRIX PART A: INTRO Every square matrix consisting of scalars (for example, real numbers) has a determinant, denoted by det A ( ) or A , which is also a scalar. PART B: SHORTCUTS FOR COMPUTING DETERMINANTS (We will discuss a general method in Part C . The shortcuts described here for small matrices may be derived from that method.) 1 × 1 Matrices If A = c , then det A ( ) = c . Warning: It may be confusing to write A = c . Don’t confuse determinants (which can be negative in value) with absolute values (which cannot ). 2 × 2 Matrices (“Butterfly Rule”) If A = a b c d , then det A ( ) = ad bc . i.e., a b c d = ad bc . (Brackets are typically left out.) We discussed this case in Section 8.3: Notes 8.55 .
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(Section 8.4: The Determinant of a Square Matrix) 8.58 3 × 3 Matrices (“Sarrus’s Rule,” named after George Sarrus) If A is 3 × 3 , then, to find det A ( ) : 1) Rewrite the 1 st and 2 nd columns on the right (as “Columns 4 and 5”). 2) Add the products along the three full diagonals that extend from upper left to lower right. 3) Subtract the products along the three full diagonals that extend from lower left to upper right. The wording above is admittedly awkward. Look at this Example: Example Let A = - 1 1 - 2 3 2 1 0 - 1 - 1 . Find det A ( ) . i.e., Find - 1 1 - 2 3 2 1 0 - 1 - 1 . Solution We begin by rewriting the 1 st and 2 nd columns on the right. - 1 1 - 2 3 2 1 0 - 1 - 1 - 1 1 3 2 0 - 1
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(Section 8.4: The Determinant of a Square Matrix) 8.59 In order to avoid massive confusion with signs, we will set up a template that clearly indicates the products that we will add and those that we will subtract. The “product along a [full] diagonal” is obtained by multiplying together the three numbers that lie along the diagonal. We will compute the six products corresponding to our six indicated diagonals, place them in the parentheses in our template, and compute the determinant. Time-Saver: If a diagonal contains a “0,” then the corresponding
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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M1410804 - (Section 8.4: The Determinant of a Square...

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