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Chapter 12: Problem 2

Given $$ A=\left(\begin{array}{ll} 6 & 1 \\ 3 & 7 \end{array}\right) $$ state (a) \(A-2 I_{2}\) (b) \(A-\lambda I_{2}\) where \(\lambda\) is a constant.

Short Answer

Expert verified
(b) \(A-\lambda I_{2}\) for a given constant \(\lambda\)? Answer: (a) The matrix \(A-2 I_{2}\) is \(\begin{pmatrix} 4 & 1 \\ 3 & 5 \end{pmatrix}\). (b) The matrix \(A-\lambda I_{2}\) is \(\begin{pmatrix} 6-\lambda & 1 \\ 3 & 7-\lambda \end{pmatrix}\), where \(\lambda\) is a constant.

Step by step solution

01

Find the 2x2 identity matrix

The identity matrix is a square matrix that has 1s on its main diagonal (from the top left to the bottom right) and 0s elsewhere. The 2x2 identity matrix, denoted as \(I_2\), can be written as: $$ I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$
02

Calculate \(A-2I_2\)

Next, we will multiply the identity matrix by 2 and then subtract it from matrix A: $$ A-2I_2 = \begin{pmatrix} 6 & 1 \\ 3 & 7 \end{pmatrix} - 2\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$ Subtracting the corresponding elements in the matrices, we get: $$ A-2I_2 = \begin{pmatrix} 6-2(1) & 1-2(0) \\ 3-2(0) & 7-2(1) \end{pmatrix} = \begin{pmatrix} 4 & 1 \\ 3 & 5 \end{pmatrix} $$
03

Calculate \(A-\lambda I_2\)

Now, we will subtract \(\lambda I_2\) from matrix A, where \(\lambda\) is a constant: $$ A-\lambda I_2 = \begin{pmatrix} 6 & 1 \\ 3 & 7 \end{pmatrix} - \lambda\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$ Subtracting the corresponding elements in the matrices, we get: $$ A-\lambda I_2 = \begin{pmatrix} 6-\lambda(1) & 1-\lambda(0) \\ 3-\lambda(0) & 7-\lambda(1) \end{pmatrix} = \begin{pmatrix} 6-\lambda & 1 \\ 3 & 7-\lambda \end{pmatrix} $$ Now, we have found both matrices: (a) \(A-2 I_{2}=\begin{pmatrix} 4 & 1 \\ 3 & 5 \end{pmatrix}\) (b) \(A-\lambda I_{2}=\begin{pmatrix} 6-\lambda & 1 \\ 3 & 7-\lambda \end{pmatrix}\) where \(\lambda\) is a constant.

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Most popular questions from this chapter

Refer to matrices \(A, B\) and \(C\) where $$ A=\left(\begin{array}{cc} 3 & 1 \\ -1 & 2 \end{array}\right), \quad B=\left(\begin{array}{ccc} 1 & 4 & 0 \\ 2 & 7 & -1 \end{array}\right), \quad C=\left(\begin{array}{cc} -2 & 1 \\ 4 & -1 \\ 0 & 3 \end{array}\right) $$ Calculate the following products. \(A B\)

\(B^{\mathrm{T}} A C^{\mathrm{T}}\)

The matrix, \(H\), is defined by $$ H=\left(\begin{array}{cc} 2 & 1 \\ \alpha & 0 \\ -3 & \beta \end{array}\right) $$ (a) State the size of \(H\). (b) State \(h_{11}, h_{21}, h_{32}\)

State the transpose of \(C\) where $$ C=\left(\begin{array}{cc} 9 & 3 \\ 1 & -2 \\ \alpha & 4 \end{array}\right) $$

Refer to matrices \(P, Q\) and \(R\) where \(P\) is a \(3 \times 2\) matrix, \(Q\) is a \(3 \times 3\) matrix and \(R\) is a \(2 \times 3\) matrix. State the size of each of the following: (a) \(P^{\mathrm{T}^{7}}\) (b) \(Q^{\mathrm{T}}\) (c) \(R^{\mathrm{T}}\) (d) \(R^{T} P^{\mathrm{T}}\) (e) \(P^{\mathrm{T}} Q^{\mathrm{T}}\)
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