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Chapter 1: Q31Q (page 1)

Show that \({I_m}A = A\) when \(A\) is an \(m \times n\) matrix. You can assume \({I_m}x = x\) for all \(x\) in \({\mathbb{R}^{\bf{m}}}\).

Short Answer

Expert verified

The matrix equation \({I_m}A = A\) is shown.

Step by step solution

01

Write A in the matrix form

The matrix form of A is\(A = \left( {\begin{aligned}{*{20}{c}}{{a_1}}&{{a_2}}&{...}&{{a_n}}\end{aligned}} \right)\).

Here, \({a_i} \in {\mathbb{R}^m}\) for all \(i = 1,2,...,n\).

02

Compute the matrix \({I_m}A\)

\(\begin{aligned}{c}{I_m}A = {I_m}\left( {\begin{aligned}{*{20}{c}}{{a_1}}&{{a_2}}&{...}&{{a_n}}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{{I_m}{a_1}}&{{I_m}{a_2}}&{...}&{{I_m}{a_n}}\end{aligned}} \right)\end{aligned}\)

Use \({I_m}{a_i} = {a_i}\) for all \(i = 1,2,...,n\).

03

Draw a conclusion\(\)

\({I_m}x\)reduces to \({I_m}A = \left( {\begin{aligned}{*{20}{c}}{{a_1}}&{{a_2}}&{...}&{{a_n}}\end{aligned}} \right) = A\).

Hence, \({I_m}A = A\) is shown.

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Most popular questions from this chapter

Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.

17. \(\left[ {\begin{array}{*{20}{c}}2&3&h\\4&6&7\end{array}} \right]\)

Construct a \(2 \times 3\) matrix \(A\), not in echelon form, such that the solution of \(Ax = 0\) is a plane in \({\mathbb{R}^3}\).

In Exercise 2, compute \(u + v\) and \(u - 2v\).

2. \(u = \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right]\), \(v = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\).

Give a geometric description of span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}8\\2\\{ - 6}\end{array}} \right]\) and \({{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{12}\\3\\{ - 9}\end{array}} \right]\).

Give a geometric description of Span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors in Exercise 16.
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