Determinants of Elementary Matrices Find the determinants for each of the elementary matrices (Sec. 3.3, Problems 40 and 41) formed by the elementary row operations on the 3 × 3 identity matrix I, described as follows.
(a) Interchange two rows of I to get matrix EInt. Find |EInt|.
(b) Replace a row by the sum of a multiple of another row and the original row to obtain matrix ERepl. Find |ERepl|.
(c) Scale a row by multiplying by a nonzero scalar k to obtain EScale. Find |EScale|.
The conclusions all extend to n × n matrices, and in combination with Problem will prove the rules for the effects of row operations on determinants.
Problem
Determinant of a Product Complete the proof that the determinant of a product is the product of the determinants, using the results of the previous problem and, from Sec. 3.3, Problems 40 and 41.
(a) Show that if A is not invertible, then |AB| = |A| |B|. HINT: By Problem 34 in Sec. 3.3, if A is not invertible, then neither is AB.
(b) When A is invertible, you should use the fact that
AB = (EpEp−1 ··· E1I)B,
where each Ej represents an elementary matrix for a row operation, to show that |AB| = |A| |B|. HINT: First show that |AB| = (−1)s k1k2 ··· ks |B| for some integer s and constants k1, k2, …, ks.
Problem 40
Elementary Matrices If we perform a single row operation on an identity matrix, we obtain an elementary matrix EInt, ERepl, or EScale. Find the elementary matrices for each of the following row operations on I3.
(a) Interchange rows 1 and 2 (EInt).
(b) Add k times row 1 to row 3 (ERepl).
(c) Multiply k times row 2 (EScale).
Problem 41
Invertibility of Elementary Matrices Explain why all elementary matrices must be invertible. Demonstrate this property by finding the inverses of EInt, ERepl, or EScale in Problem.
Problem
Elementary Matrices If we perform a single row operation on an identity matrix, we obtain an elementary matrix EInt, ERepl, or EScale. Find the elementary matrices for each of the following row operations on I3.
(a) Interchange rows 1 and 2 (EInt).
(b) Add k times row 1 to row 3 (ERepl).
(c) Multiply k times row 2 (EScale).
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