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Chapter 2: Problem 70

Write \(A=\left[\begin{array}{ll}4 & 5 \\ 1 & 3\end{array}\right]\) as the sum of a symmetric matrix \(B\) and a skew-symmetric matrix \(C\).

Short Answer

Expert verified
The symmetric matrix \(B\) is: \[B = \left[\begin{array}{ll}4 & 3 \\3 & 3\end{array}\right]\] and the skew-symmetric matrix \(C\) is: \[C = \left[\begin{array}{ll}0 & 2 \\-2 & 0\end{array}\right]\]

Step by step solution

01

Initialize Matrix A

Given the matrix A: \[A = \left[\begin{array}{ll}4 & 5 \\1 & 3\end{array}\right]\] Step 2: Analyze and Find Matrix B
02

Analyze and Find Matrix B

To find symmetric matrix B, analyze each element in matrix A and create a new matrix with the average of the corresponding elements and their transposed counterpart. \[B = \frac{1}{2}(A + A^T)\] Calculate the transpose of A: \[A^T = \left[\begin{array}{ll}4 & 1 \\5 & 3\end{array}\right]\] Then, calculate B: \[B = \frac{1}{2}(\left[\begin{array}{ll}4 & 5 \\1 & 3\end{array}\right] + \left[\begin{array}{ll}4 & 1 \\5 & 3\end{array}\right])\\ = \frac{1}{2}\left[\begin{array}{ll}8 & 6 \\6 & 6\end{array}\right] = \left[\begin{array}{ll}4 & 3 \\3 & 3\end{array}\right]\] Step 3: Analyze and Find Matrix C
03

Analyze and Find Matrix C

To find skew-symmetric matrix C, analyze each element in matrix A and create a new matrix with the difference of the corresponding elements and their transposed counterpart. \[C = \frac{1}{2}(A - A^T)\] Using the transpose of A calculated earlier: \[C = \frac{1}{2}(\left[\begin{array}{ll}4 & 5 \\1 & 3\end{array}\right] - \left[\begin{array}{ll}4 & 1 \\5 & 3\end{array}\right])\\ = \frac{1}{2}\left[\begin{array}{ll}0 & 4 \\-4 & 0\end{array}\right] = \left[\begin{array}{ll}0 & 2 \\-2 & 0\end{array}\right]\] Step 4: Verify the sum
04

Verify the sum

Check if the sum of symmetric matrix B and skew-symmetric matrix C equals the given Matrix A: \[B + C = \left[\begin{array}{ll}4 & 3 \\3 &3\end{array}\right] + \left[\begin{array}{ll}0 & 2 \\-2 & 0\end{array}\right] = \left[\begin{array}{ll}4 & 5 \\1 & 3\end{array}\right]\] As we can see, the sum of B and C equals the original matrix A: \[A = B + C\] Therefore, the symmetric matrix B is: \[B = \left[\begin{array}{ll}4 & 3 \\3 & 3\end{array}\right]\] and the skew-symmetric matrix C is: \[C = \left[\begin{array}{ll}0 & 2 \\-2 & 0\end{array}\right]\]

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Most popular questions from this chapter

Suppose \(A B=C,\) where \(A\) and \(C\) are upper triangular. (a) Find \(2 \times 2\) nonzero matrices \(A, B, C,\) where \(B\) is not upper triangular. (b) Suppose \(A\) is also invertible. Show that \(B\) must also be upper triangular.

Let \(V\) and \(W\) be vector spaces with subspaces \(V_{1}\) and \(W_{1}\), respectively. If \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) is linear, prove that \(\mathrm{T}\left(\mathrm{V}_{1}\right)\) is a subspace of \(\mathrm{W}\) and that $\left\\{x \in \mathrm{V}: \mathrm{T}(x) \in \mathrm{W}_{1}\right\\}\( is a subspace of \)\mathrm{V}$.

Suppose \(A\) is a square matrix. Show (a) \(A+A^{T}\) is symmetric, (b) \(A-A^{T}\) is skew-symmetric, (c) \(A=B+C,\) where \(B\) is symmetric and \(C\) is skew- symmetric.

Label the following statements as true or false. Assume that \(V\) and \(\mathrm{W}\) are finite-dimensional vector spaces with ordered bases \(\beta\) and \(\gamma\), respectively, and $\mathrm{T}, \mathrm{U}: \mathrm{V} \rightarrow \mathrm{W}$ are linear transformations. (a) For any scalar \(a, a \mathbf{T}+\mathbf{U}\) is a linear transformation from \(\mathbf{V}\) to \(\mathbf{W}\). (b) \([\mathrm{T}]_{\beta}^{\gamma}=[\mathrm{U}]_{\beta}^{\gamma}\) implies that \(\mathrm{T}=\mathrm{U}\). (c) If \(m=\operatorname{dim}(\mathrm{V})\) and \(n=\operatorname{dim}(\mathrm{W})\), then \([\mathrm{T}]_{\beta}^{\gamma}\) is an \(m \times n\) matrix. (d) \([\mathrm{T}+\mathrm{U}]_{\beta}^{\gamma}=[\mathrm{T}]_{\beta}^{\gamma}+[\mathrm{U}]_{\beta}^{\gamma}\). (e) \(\mathcal{L}(\mathrm{V}, \mathrm{W})\) is a vector space. (f) \(\mathcal{L}(\mathrm{V}, \mathrm{W})=\mathcal{L}(\mathrm{W}, \mathrm{V})\).

Refer to the following matrices: $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 3 & 4 \end{array}\right], \quad B=\left[\begin{array}{rrr} 4 & 0 & -3 \\ -1 & -2 & 3 \end{array}\right], \quad C=\left[\begin{array}{rrrr} 2 & -3 & 0 & 1 \\ 5 & -1 & -4 & 2 \\ -1 & 0 & 0 & 3 \end{array}\right], \quad D=\left[\begin{array}{rr} 2 \\ -1 \\ 3 \end{array}\right].$$ Find (a) \(3 A-4 B,\) (b) \(A C,\) (c) \(B C,\) (d) \(A D,\) (e) \(B D,\) (f) \(C D.\)
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