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Chapter 4: Problem 2

Compute the determinants of the following matrices in $\mathrm{M}_{2 \times 2}(R)$. (a) \(\left(\begin{array}{rr}6 & -3 \\ 2 & 4\end{array}\right)\) (b) \(\left(\begin{array}{rr}-5 & 2 \\ 6 & 1\end{array}\right)\) (c) \(\left(\begin{array}{rr}8 & 0 \\ 3 & -1\end{array}\right)\)

Short Answer

Expert verified
The determinants of the given matrices are: (a) \(\det\left(\begin{array}{rr}6 & -3 \\\ 2 & 4\end{array}\right) = 30\) (b) \(\det\left(\begin{array}{rr}-5 & 2 \\\ 6 & 1\end{array}\right) = -17\) (c) \(\det\left(\begin{array}{rr}8 & 0 \\\ 3 & -1\end{array}\right) = -8\)

Step by step solution

01

Calculate the determinant of matrix (a)

To calculate the determinant of the matrix \(\left(\begin{array}{rr}6 & -3 \\\ 2 & 4\end{array}\right)\), we will apply the formula \(\det(A) = ad - bc\). In this case, \(a = 6\), \(b = -3\), \(c = 2\), and \(d = 4\). Therefore, \[\det(A) = (6)(4) - (-3)(2) = 24 + 6 = 30.\]
02

Calculate the determinant of matrix (b)

To calculate the determinant of the matrix \(\left(\begin{array}{rr}-5 & 2 \\\ 6 & 1\end{array}\right)\), we will apply the formula \(\det(A) = ad - bc\). In this case, \(a = -5\), \(b = 2\), \(c = 6\), and \(d = 1\). Therefore, \[\det(A) = (-5)(1) - (2)(6) = -5 - 12 = -17.\]
03

Calculate the determinant of matrix (c)

To calculate the determinant of the matrix \(\left(\begin{array}{rr}8 & 0 \\\ 3 & -1\end{array}\right)\), we will apply the formula \(\det(A) = ad - bc\). In this case, \(a = 8\), \(b = 0\), \(c = 3\), and \(d = -1\). Therefore, \[\det(A) = (8)(-1) - (0)(3) = -8.\] To summarize, the determinants of the given matrices are: (a) \(\det\left(\begin{array}{rr}6 & -3 \\\ 2 & 4\end{array}\right) = 30\) (b) \(\det\left(\begin{array}{rr}-5 & 2 \\\ 6 & 1\end{array}\right) = -17\) (c) \(\det\left(\begin{array}{rr}8 & 0 \\\ 3 & -1\end{array}\right) = -8\)

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Most popular questions from this chapter

Consider the vector space \(\mathbf{P}_{3}(t)\) of polynomials of degree \(\leq 3\). (a) Show that \(S=\left\\{(t-1)^{3},(t-1)^{2}, t-1,1\right\\}\) is a basis of \(\mathbf{P}_{3}(t)\), (b) Find the coordinate vector \([v]\) of \(v=3 t^{3}-4 t^{2}+2 t-5\) relative to \(S\).

Suppose \(u\) and \(v\) belong to a vector space \(V\). Simplify each of the following expressions: (a) \(E_{1}=4(5 u-6 v)+2(3 u+v)\), (c) \(\quad E_{3}=6(3 u+2 v)+5 u-7 v\), (b) \(E_{2}=5(2 u-3 v)+4(7 v+8)\), (d) \(E_{4}=3(5 u+2 / v)\).

Prove that an upper triangular \(n \times n\) matrix is invertible if and only if all its diagonal entries are nonzero.

Let \(A\) and \(B\) be arbitrary \(m \times n\) matrices. Show that \(\operatorname{rank}(A+B) \leq \operatorname{rank}(A)+\operatorname{rank}(B)\).

Suppose that \(A_{1}, A_{2}, \ldots\) are linearly independent sets of vectors, and that \(A_{1} \subseteq A_{2} \subseteq \ldots .\) Show that the union \(A=A_{1} \cup A_{2} \cup \ldots\) is also linearly independent.
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