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Chapter 3: Problem 81

Comment on the following claim: Every matrix equation represents a system of equations.

Short Answer

Expert verified
The claim is correct. Every matrix equation of the form AX = B indeed represents a system of linear equations. In our example, the matrix equation represented a system of 2 linear equations with 2 unknowns (x_1 and x_2), and this can be generalized to larger matrices as well.

Step by step solution

01

Define a matrix equation with variables

Let's consider a matrix equation of the form AX = B, where A is a matrix of size m x n, X is a column matrix of size n x 1 (or just n, 1 column vector), and B is a column matrix of size m x 1. The matrices A and B have known values, and X contains the unknown variables. Example: Given A = \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), X = \(\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\), and B = \(\begin{bmatrix} e \\ f \end{bmatrix}\). The matrix equation AX = B becomes: \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) \(\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\) = \(\begin{bmatrix} e \\ f \end{bmatrix}\)
02

Illustrate how the matrix equation represents a system of equations

Now, let's multiply the matrices to obtain the following result: \(\begin{bmatrix} ax_1 + bx_2 \\ cx_1 + dx_2 \end{bmatrix}\) = \(\begin{bmatrix} e \\ f \end{bmatrix}\) We can observe that the elements of the resulting matrix equate to the elements of the matrix B: \(ax_1 + bx_2 = e\) \(cx_1 + dx_2 = f\) This gives us a system of linear equations: 1. \(ax_1 + bx_2 = e\) 2. \(cx_1 + dx_2 = f\)
03

Comment on the claim

In conclusion, the given claim is correct. Every matrix equation, specifically of the form AX = B, does represent a system of linear equations. In our example with a 2x2 matrix, the matrix equation represented a system of 2 linear equations with 2 unknowns (x_1 and x_2). This can be generalized to larger matrices as well.

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