## Linear Algebra and Its Applications, exercise 1.4.12

Exercise 1.4.12. Given the matrices A and B below, the first row of their product AB is a linear combination of all the rows of B. Find the coefficients of this linear combination, and the first row of B.

$A = \begin{bmatrix} 2&1&4 \\ 0&-1&1 \end{bmatrix} \quad B = \begin{bmatrix} 1&1 \\ 0&1 \\ 1&0 \end{bmatrix}$

Answer: Since A is a 3×2 matrix and B is a 2×3 matrix, their product AB is a 2×2 matrix. In computing the first row of B we use the first row of A, with the three values 2, 1, and 4 respectively being the coefficients of the linear combination:

$2 \cdot \begin{bmatrix} 1&1 \end{bmatrix} + 1 \cdot \begin{bmatrix} 0&1 \end{bmatrix} + 4 \cdot \begin{bmatrix} 1&0 \end{bmatrix} = \begin{bmatrix} 6&3 \end{bmatrix}$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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