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Chapter 2: Problem 16

Find (a) \(A B\) and (b) \(B A\) (if they are defined). $$A=\left[\begin{array}{r} 6 \\ -2 \\ 1 \\ 6 \end{array}\right], \quad B=\left[\begin{array}{ll} 10 & 12 \end{array}\right]$$

Short Answer

Expert verified
a) \(AB =\[[36, 36]\]\) b) \(BA\) is not defined

Step by step solution

01

Analyze Matrices for Multiplication

First, we need to determine whether the multiplication of matrices A and B is possible in both sequences (AB and BA). We can do this by examining their structure - Matrix A is a 4x1 matrix (4 rows and 1 column), and Matrix B is a 1x2 matrix (1 row and 2 columns).
02

Matrix Multiplication: Finding \(AB\)

We can start with finding \(AB\). Matrix multiplication is possible here because the number of columns in A (which is 1) is equal to the number of rows in B (which is also 1). We multiply the corresponding elements and add the results: \(AB = A[0]*B[0] + A[1]*B[1] = 6*10 + -2*12 + 1*0 + 6*0 = 60 -24 +0 + 0 = 36\). Hence, \(AB =\[[36, 36]\]\)
03

Matrix Multiplication: Attempting to Find \(BA\)

Next, let's attempt to find \(BA\). In this case, matrix multiplication is not possible because the number of columns in B (which is 2) is not equal to number of rows in A (which is 4). Thus, \(BA\) does not exist.

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Most popular questions from this chapter

Let \(P\) be a \(2 \times 2\) stochastic matrix. Prove that there exists a \(2 \times 1\) state matrix \(X\) with nonnegative entries such that \(P X=X\)

find the \(L U\) -factorization of the matrix. $$\left[\begin{array}{rrr} 3 & 0 & 1 \\ 6 & 1 & 1 \\ -3 & 1 & 0 \end{array}\right]$$

Determine whether the matrix is idempotent. A square matrix \(A\) is idempotent if \(A^{2}=A\) $$\left[\begin{array}{rr} 2 & 3 \\ -1 & -2 \end{array}\right]$$

Prove that if the matrix \(I-A B\) is nonsingular, then so is \(I-B A\)

Factor the matrix \(A\) into a product of elementary matrices. $$A=\left[\begin{array}{rrr} 1 & -2 & 0 \\ -1 & 3 & 0 \\ 0 & 0 & 1 \end{array}\right]$$
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