In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 A = [1 2 3 4], B = 3 4

Verified step by step guidance

1. Step 1: Identify the dimensions of matrices A and B. Matrix A is a 1x4 matrix (1 row, 4 columns), and matrix B is a 4x1 matrix (4 rows, 1 column).
2. Step 2: To find the product AB, check if the number of columns in A equals the number of rows in B. Since A is 1x4 and B is 4x1, multiplication AB is possible.
3. Step 3: Multiply A and B by taking the dot product of the single row of A with the single column of B. This results in a 1x1 matrix (a scalar). The formula is: $\mathrm{AB}=\; [1\; imes\; 1\; +\; 2\; imes\; 2\; +\; 3\; imes\; 3\; +\; 4\; imes\; 4]$.
4. Step 4: To find the product BA, check if the number of columns in B equals the number of rows in A. B is 4x1 and A is 1x4, so multiplication BA is possible.
5. Step 5: Multiply B and A by taking the outer product of the 4x1 matrix B and the 1x4 matrix A. This results in a 4x4 matrix where each element is the product of the corresponding elements from B and A.

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. The number of columns in the first matrix must equal the number of rows in the second matrix for the product to be defined.

Matrix Dimensions and Compatibility

The dimensions of a matrix are given as rows × columns. For two matrices A and B, the product AB is defined only if the number of columns in A equals the number of rows in B. This concept determines whether AB or BA can be computed.

Row and Column Vectors

A row vector is a 1 × n matrix, and a column vector is an n × 1 matrix. Multiplying a row vector by a column vector results in a scalar, while multiplying a column vector by a row vector results in a matrix. Understanding this helps in computing AB and BA.