2.4.6. When computing the determinant of a matrix by hand, it's common to use cofactor expansion and apply the definition recursively. But this is terribly inefficient as a function of the matrix size. (a) Explain why, if A = LU is an LU factorization, det(A) = U11U22 ··· Unn = ?(i=1 to n) Uii. (b) Using the result of part (a), write a function determinant(A) that computes the determinant using Function 2.4.1. Use your function and the built-in det on the matrices magic(n) for n = 3, 4, ..., 7, and make a table showing n, the value from your function, and the relative error when compared to det. (c) Show that determinant fails for magic(8) but is fine for magic(9). Speculate on what property of these two matrices makes the results so different.
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Hint: In this exercise you have to use the following property: if a matrix V is obtained from a matrix U by multiplying 1 row or 1 column of U by a scalar k, then det(V) = k det(U). Let M be the matrix Complete the following: a) The determinant of M can be expressed as 5det A, where A is the 3*3 matrix defined as (Matrix A to be completed). b) The determinant of A can be expressed as 3det B,where B is the 2*2 matrix defined as (Matrix B to be completed) c) The determinant of B is d) Thus the determinant of M is
Adi S.
The determinant of a $3 \times 3$ matrix $A$ is defined as follows. $$\begin{array}{l}\text { If } A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{array}\right] \text { , then } \\\qquad|A|=\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{array}\right|=\left(a_{11} a_{22} a_{33}+a_{12} a_{23}a_{31}+a_{13}a_{21} a_{32}\right) \\-\left(a_{31} a_{22} a_{13}+a_{32} a_{23} a_{11}+a_{33} a_{21} a_{12}\right)\end{array}$$ Refer to Exercise $32 .$ Find the determinant by expanding about column 1 and using the method of cofactors. Do these methods give the same determinant for $3 \times 3$ matrices?
Systems and Matrices
Determinant Solution of Linear Systems
The determinant of a $3 \times 3$ matrix $A$ is defined as follows. $$\text {If } A=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right], \text {then }|A|=\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|$$ $$\begin{aligned} =&\left(a_{11} a_{22} a_{33}+a_{12} a_{23} a_{31}+a_{13} a_{21} a_{32}\right) \\ &-\left(a_{31} a_{22} a_{13}+a_{32} a_{23} a_{11}+a_{33} a_{21} a_{12}\right) \end{aligned}$$ Work these exercises in order. Does the method of evaluating a determinant using "diagonals" extend to $4 \times 4$ matrices?
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