Linear Algebra

Topic: Cramer’s Rule 325Topic: Cramer’s RuleWe have introduced determinant functions algebraically by looking for a formulato decide whether a matrix is nonsingular. After that introduction we saw ageometric interpretation, that the determinant function gives the size of the boxwith sides formed by the columns of the matrix. This Topic makes a connectionbetween the two views.First, a linear systemx 1 + 2x 2 = 63x 1 + x 2 = 8is equivalent to a linear relationship among vectors.( ( (1 2 6x 1 · + x3)2 · =1)8)The picture below shows a parallelogram with sides formed from ( (13)and2)( 1nested inside a parallelogram with sides formed from x 1(1 3)and2x21).( 68)x 1 ·( 13)( 13)( 21) x 2 · ( 21)So even without determinants we can state the algebraic issue that opened thisbook, finding the solution of a linear system, in geometric terms: by what factorsx 1 and x 2 must we dilate the vectors to expand the small parallegram to fill thelarger one?However, by employing the geometric significance of determinants we canget something that is not just a restatement, but also gives us a new insight andsometimes allows us to compute answers quickly. Compare the sizes of theseshaded boxes.( 68)x 1 ·( 13)( 13)( 21)( 21)( 21)The second is formed from x 1( 13)and( 21), and one of the properties of the sizefunction — the determinant — is that its size is therefore x 1 times the size of the