## 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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