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Question Video: Finding the Number of Solutions to a System of Linear Equations Using Matrix Ranks Mathematics

Find the number of solutions for the following system of linear equations: [−5, −1, −11 and −2, 14, −10 and 14, −3, 12][𝑥 and 𝑦 and 𝑧] = [−10 and −4 and −7].

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Video Transcript

Find the number of solutions for the following system of linear equations.

Recall that the Rouché–Capelli theorem states that solutions to the system of linear equations exist if, and only if, the rank of its coefficient matrix 𝐴 is equal to the rank of its augmented matrix 𝐴 stroke 𝑏. Consider the coefficient matrix 𝐴. It is clearly not the zero matrix, and it contains no rows or columns that are scalar multiples of each other. Taking the determinant of 𝐴 by expanding along the top row, we get a result of 1516. We have therefore found a three-by-three submatrix of 𝐴 with a nonzero determinant. Therefore, the rank of 𝐴 is three.

Now consider the augmented matrix 𝐴 stroke 𝑏. This is a three-by-four matrix. And recall that the rank of a matrix cannot exceed the minimum of either the number of rows or columns. Therefore, the rank of this matrix must be at most three. By construction, 𝐴 stroke 𝑏 contains 𝐴 as a submatrix, which we have just shown to have a nonzero determinant. Therefore, the rank of 𝐴 stroke 𝑏 must be at least three and at most three. And therefore, the rank of 𝐴 stroke 𝑏 is equal to three.

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