Determine whether the product A B or B A is defined. If a...

Determine whether the product $A B$ or $B A$ is defined. If a product is defined, state its size ( number of rows and columns). Do not actually calculate any products. $$A=\left(\begin{array}{rr} -4 & 15 \\ 3 & -7 \\ 2 & 10 \end{array}\right), \quad B=\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right)$$

Determine whether the product $A B$ or $B A$ is defined. If a product is defined, state its size ( number of rows and columns). Do not actually calculate any products.
$$A=\left(\begin{array}{rr}
-4 & 15 \\
3 & -7 \\
2 & 10
\end{array}\right), \quad B=\left(\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right)$$

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Key Concepts

Dimension Determination

Once the conformability condition is met, the dimensions of the resulting matrix are defined by the number of rows of the first matrix and the number of columns of the second matrix. This concept is crucial in understanding and predicting the size of the product matrix without performing the element-by-element computation.

Matrix Multiplication

Matrix multiplication involves combining two matrices by taking the dot product of rows of the first matrix with columns of the second. This operation is not commutative, meaning that the order of multiplication matters, which impacts whether a product is defined and what the resulting matrix will be.

Conformability Conditions

For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. This condition, often referred to as conformability, is essential to determine if the multiplication operation is valid.

Transcript

00:01 Okay, we want to know whether the product ab or ba is defined or undefined.

00:10 Okay, so the first thing we need to do is find the size of matrix a and the size of matrix b.

00:17 So we need to count the rows and the columns.

00:20 So one, two, three.

00:23 Okay, three rows, one by two columns.

00:30 So three rows by two columns means the size of matrix.

00:33 Matrix a is 3 by 2.

00:36 Okay, let's try b.

00:38 1, 2.

00:40 It's two rows.

00:44 1, 2.

00:45 It's two columns.

00:47 Okay? so matrix b is a 2 by 2 matrix.

00:51 All right.

00:52 So now, since order matters, a times b is not necessarily the same thing as b times a.

00:59 Okay, so we need to do this twice.

01:01 So first we're going to see if a, b is defined or undefined.

01:06 So we need to take matrix a, the dimensions of matrix a, so three by two, and then we need to take the dimensions of matrix b, which is two by two.

01:20 And then first we're going to look at the two numbers on the inside.

01:24 Okay, so you have a two here and a two here...

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