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Chapter 4: Problem 2

Evaluate the determinant of the following \(2 \times 2\) matrices. (a) \(\left(\begin{array}{rr}4 & -5 \\ 2 & 3\end{array}\right)\) (b) \(\left(\begin{array}{rr}-1 & 7 \\ 3 & 8\end{array}\right)\) (c) \(\left(\begin{array}{cc}2+i & -1+3 i \\ 1-2 i & 3-i\end{array}\right)\) (d) \(\left(\begin{array}{cc}3 & 4 i \\ -6 i & 2 i\end{array}\right)\)

Short Answer

Expert verified
The determinants of the given matrices are: (a) 22 (b) -29 (c) \(10 - 5i\) (d) \(-24 + 6i\)

Step by step solution

01

Identify the elements

We need to identify the elements of the matrix as follows: \(a = 4, b = -5, c = 2, d = 3\).
02

Apply the determinant formula

The determinant is given by \(ad - bc = (4)(3) - (-5)(2) = 12 + 10 = 22\). (b) For the matrix \(\left(\begin{array}{rr}-1 & 7\\ 3 & 8\end{array}\right)\):
03

Identify the elements

We need to identify the elements of the matrix as follows: \(a = -1, b = 7, c = 3, d = 8\).
04

Apply the determinant formula

The determinant is given by \(ad - bc = (-1)(8) - (7)(3) = -8 - 21 = -29\). (c) For the matrix \(\left(\begin{array}{cc}2+i & -1+3 i\\ 1-2 i & 3-i\end{array}\right)\):
05

Identify the elements

We need to identify the elements of the matrix as follows: \(a = 2+i, b = -1+3i, c = 1-2i, d = 3-i\).
06

Apply the determinant formula

The determinant is given by \(ad - bc = (2+i)(3-i) - (-1+3i)(1-2i) = [(2+i)(3-i) +(-1+3i)(1-2i)] = (6-2i+i^2) +(-1+2i+3i-6i^2) = (6-2i-1) -(1-5i+6) = 5-2i -(-5+5i) = 10-5i\). (d) For the matrix \(\left(\begin{array}{cc}3 & 4 i\\ -6 i & 2 i\end{array}\right)\):
07

Identify the elements

We need to identify the elements of the matrix as follows: \(a = 3, b = 4i, c = -6i, d = 2i\).
08

Apply the determinant formula

The determinant is given by \(ad - bc = (3)(2i) - (4i)(-6i) = 6i -(-24i^2) = 6i + 24(-1) = -24 + 6i\).

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Most popular questions from this chapter

Prove that if \(M \in M_{n \times n}(F)\) can be written in the form $$ M=\left(\begin{array}{ll} A & B \\ O & C \end{array}\right) $$ where \(A\) and \(C\) are square matrices, then \(\operatorname{det}(M)=\operatorname{det}(A) \cdot \operatorname{det}(C)\). Visit goo.g1/pgMdpX for a solution.

For \(k=1,2, \ldots, 5,\) find the number \(n_{k}\) of linearly independent subsets consisting of \(k\) columns for each of the following matrices: (a) \(A=\left[\begin{array}{lllll}1 & 1 & 0 & 2 & 3 \\ 1 & 2 & 0 & 2 & 5 \\ 1 & 3 & 0 & 2 & 7\end{array}\right]\) (b) \(B=\left[\begin{array}{lllll}1 & 2 & 1 & 0 & 2 \\ 1 & 2 & 3 & 0 & 4 \\ 1 & 1 & 5 & 0 & 6\end{array}\right]\)

Find the rank of cach of the following matrices: (a) \(\left[\begin{array}{rrrrr}1 & 3 & -2 & 5 & 4 \\ 1 & 4 & 1 & 3 & 5 \\ 1 & 4 & 2 & 4 & 3 \\ 2 & 7 & -3 & 6 & 13\end{array}\right]\) (b) \(\left[\begin{array}{rrrr}1 & 2 & -3 & -2 \\ 1 & 3 & -2 & 0 \\ 3 & 8 & -7 & -2 \\ 2 & 1 & -9 & -10\end{array}\right]\) (c) \(\left[\begin{array}{rrr}1 & 1 & 2 \\ 4 & 5 & 5 \\ 5 & 8 & 1 \\ -1 & -2 & 2\end{array}\right]\)

Let \(V\) be the vector space of \(n\) -square matrices. Let \(U\) be the subspace of upper triangular matrices, and let \(W\) be the subspace of lower triangular matrices. Find (a) \(U \cap W\), (b) \(U+W\).

Let \(r=\operatorname{rank}(A+B) .\) Find \(2 \times 2\) matrices \(A\) and \(B\) such that (a) \(r < \operatorname{rank}(A), \operatorname{rank}(\mathrm{B})\); (b) \(r=\operatorname{rank}(A)=\operatorname{rank}(B)\); (c) \(r > \operatorname{rank}(A), \operatorname{rank}(\mathrm{B})\).
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