3. Use (3.1.3) O and (3.1.4) O to calculate each of the following determinants. 1 3...

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3. Use (3.1.3) O and (3.1.4) O to calculate each of the following determinants. 1 3 2 -2 4 0 -1 3 |A| =-2 1 |B| = |4 3 1 |C| = 4 1 2 -1 4 1 2 (3.1.3) A = (a11a22 a33 + a12a23a31 + a13a21 a32) - (a13a22 a31 +a11a23a32 + a12a21 a33). This gives us another way to compute A. Indeed, if we write A and adjoin it to its first two columns, we get (3.1.4) a11 а12 a13 a11 a12 a21 A22 a23 | a21 a22 аз1 A32 a33 A31 a32 The first three terms in (3.1.3) D: a11a22 a33, a12a23 a31, a13a21a32, can be obtained from (3.1.4) D in the following way: Drawing an arrow from a11 to a33 through a22 (from the top left to the bottom right), we see that the product of the three numbers a11, a22, a33 on the arrow is the first product in (3.1.3) D. Drawing an arrow from a12 to a31 through a23 and from a13 to a32 through a21, we get the other two products. Similarly, we can get the three terms a13 a22 a31, a11a23 a32, a12a21 a33 by drawing an arrow from a13 to a31 through a22 (from the top right to the bottom left), from a11 to a32 through a23 and from a12 to a33 through a21 . 3. Use (3.1.3) O and (3.1.4) O to calculate each of the following determinants. 1 3 2 -2 4 0 -1 3 |A| =-2 1 |B| = |4 3 1 |C| = 4 1 2 -1 4 1 2 (3.1.3) A = (a11a22 a33 + a12a23a31 + a13a21 a32) - (a13a22 a31 +a11a23a32 + a12a21 a33). This gives us another way to compute A. Indeed, if we write A and adjoin it to its first two columns, we get (3.1.4) a11 а12 a13 a11 a12 a21 A22 a23 | a21 a22 аз1 A32 a33 A31 a32 The first three terms in (3.1.3) D: a11a22 a33, a12a23 a31, a13a21a32, can be obtained from (3.1.4) D in the following way: Drawing an arrow from a11 to a33 through a22 (from the top left to the bottom right), we see that the product of the three numbers a11, a22, a33 on the arrow is the first product in (3.1.3) D. Drawing an arrow from a12 to a31 through a23 and from a13 to a32 through a21, we get the other two products. Similarly, we can get the three terms a13 a22 a31, a11a23 a32, a12a21 a33 by drawing an arrow from a13 to a31 through a22 (from the top right to the bottom left), from a11 to a32 through a23 and from a12 to a33 through a21 .