In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 4 2 2 3 4 A = 6 1 B = 3 5 - 1 - 2 0

Verified step by step guidance

1. Step 1: Identify the dimensions of matrices A and B. Matrix A is a 3x2 matrix (3 rows, 2 columns), and matrix B is a 2x3 matrix (2 rows, 3 columns).
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 3x2 and B is 2x3, the product AB is defined and will result in a 3x3 matrix.
3. Step 3: To find the product BA, check if the number of columns in B equals the number of rows in A. Since B is 2x3 and A is 3x2, the product BA is defined and will result in a 2x2 matrix.
4. Step 4: Calculate the product AB by multiplying each row of A by each column of B. For each element in the resulting matrix, use the formula: $c\_\mathrm{ij}=\sum k_{}{a}_{}\_\mathrm{ik}{b}_{}\_\mathrm{kj}$, where $i$ is the row index and $j$ is the column index.
5. Step 5: Calculate the product BA similarly by multiplying each row of B by each column of A, using the same summation formula for matrix multiplication.

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. The element in the ith row and jth column of the product matrix is the dot product of the ith row of the first matrix and the jth column of the second matrix.

Dimension Compatibility for Multiplication

Two matrices can be multiplied only if the number of columns in the first matrix equals the number of rows in the second. For example, if A is m×n and B is p×q, multiplication AB is defined only if n = p, resulting in an m×q matrix.

Non-Commutativity of Matrix Multiplication

Matrix multiplication is generally not commutative, meaning AB does not necessarily equal BA. Even if both products are defined, their results can differ in size and values, so both AB and BA must be computed separately.