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Chapter 2: Problem 21

Write out the diagonal matrices \(A=\operatorname{diag}(4,-3,7), B=\operatorname{diag}(2,-6), C=\operatorname{diag}(3,-8,0,5)\).

Short Answer

Expert verified
The diagonal matrices A, B, and C in their full matrix forms are: A = \(\begin{bmatrix} 4 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 7 \end{bmatrix}\), B = \(\begin{bmatrix} 2 & 0 \\ 0 & -6 \end{bmatrix}\), and C = \(\begin{bmatrix} 3 & 0 & 0 & 0 \\ 0 & -8 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 5 \end{bmatrix}\).

Step by step solution

01

Matrix A

To write out the diagonal matrix A, fill in the given diagonal elements and place 0's in all other positions. A = \(\begin{bmatrix} 4 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 7 \end{bmatrix}\)
02

Matrix B

To write out the diagonal matrix B, do the same as for A, but with the given diagonal elements of B: B = \(\begin{bmatrix} 2 & 0 \\ 0 & -6 \end{bmatrix}\)
03

Matrix C

Lastly, write out the diagonal matrix C following the same pattern with the given diagonal elements: C = \(\begin{bmatrix} 3 & 0 & 0 & 0 \\ 0 & -8 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 5 \end{bmatrix}\) Now, we have written out the diagonal matrices A, B, and C in their full matrix forms.

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Most popular questions from this chapter

Prove the following generalization of Theorem 2.23. Let $\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}$ be a linear transformation from a finite- dimensional vector space \(V\) to a finite-dimensional vector space W. Let \(\beta\) and \(\beta^{\prime}\) be ordered bases for \(\mathrm{V}\), and let \(\gamma\) and \(\gamma^{\prime}\) be ordered bases for \(\mathrm{W}\). Then $[\mathrm{T}]_{\beta}^{\gamma}^{\prime}=P^{-1}[\mathrm{~T}]_{\beta}^{\gamma} Q\(, where \)Q\( is the matrix that changes \)\beta^{\prime}$-coordinates into \(\beta\)-coordinates and \(P\) is the matrix that changes \(\gamma^{\prime}\)-coordinates into \(\gamma\)-coordinates.

Suppose \(A\) and \(B\) are symmetric. Show that the following are also symmetric: (a) \(A+B\) (b) \(k A,\) for any scalar \(k\) \((\mathrm{c}) \quad A^{2}\) (d) \(A^{n},\) for \(n>0\) (e) \(f(A),\) for any polynomial \(f(x)\)

Let \(M=\operatorname{diag}(A, B, C),\) where \(A=\left[\begin{array}{ll}1 & 2 \\\ 3 & 4\end{array}\right], B=[5], C=\left[\begin{array}{cc}1 & 3 \\ 5 & 7\end{array}\right] .\) Find \(M^{2}\).

Suppose \(A\) and \(B\) are orthogonal matrices. Show that \(A^{T}, A^{-1}, A B\) are also orthogonal.

Show (a) If \(A\) has a zero row, then \(A B\) has a zero row. (b) If \(B\) has a zero column, then \(A B\) has a zero column.
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