Can These Matrices Be Multiplied?

H1: Can These Matrices Be Multiplied?

When we talk about matrices in mathematics, one of the fundamental operations we perform is multiplication. However, not all pairs of matrices can be multiplied together. There's a specific rule, a compatibility condition, that dictates whether a matrix multiplication is possible. Let's dive into this and explore an example to make it crystal clear. Understanding this rule is crucial for anyone working with matrices, whether you're a student tackling linear algebra or a professional using mathematical models.

The Compatibility Rule for Matrix Multiplication

The core principle governing matrix multiplication is quite straightforward: for the product of two matrices, say matrix A multiplied by matrix B (A x B), to be defined, the number of columns in the first matrix (A) must be equal to the number of rows in the second matrix (B). If this condition isn't met, the multiplication is simply not possible. It's like trying to fit a square peg into a round hole – it just doesn't work.

Let's visualize this. If matrix A has dimensions $m imes n$ (meaning it has $m$ rows and $n$ columns) and matrix B has dimensions $p imes q$ (meaning it has $p$ rows and $q$ columns), then the multiplication A x B is only possible if $n = p$. If this condition holds true, the resulting matrix, C (where $C = A imes B$), will have the dimensions $m imes q$. The inner dimensions must match, and the outer dimensions will form the dimensions of the product.

Think of it this way: when you multiply matrices, each element in the resulting matrix is calculated by taking the dot product of a row from the first matrix and a column from the second matrix. For this dot product to be defined, the number of elements in the row (which is the number of columns in the first matrix) must match the number of elements in the column (which is the number of rows in the second matrix). This is why the column-row rule is so critical. Missing this fundamental step means you can't proceed with the calculation, leading to an undefined result.

Understanding this compatibility rule is not just about avoiding errors; it's about grasping the underlying structure and logic of matrix operations. It's a concept that appears in various fields, including computer graphics, data science, physics, and engineering, where matrices are used extensively to represent transformations, systems of equations, and datasets. Getting this right from the start sets a strong foundation for more complex matrix manipulations and applications.

Analyzing the Given Matrix Multiplication Options

Now, let's apply this rule to the specific options provided. We need to determine which of the following matrix multiplications is actually possible.

Option A: $\left[\begin{array}{ll}0 & 3\end{array}\right] \times\left[\begin{array}{ll}1 & -4\end{array}\right]$

First, let's determine the dimensions of the matrices in Option A. The first matrix, $\left[\begin{array}{ll}0 & 3\end{array}\right]$, has 1 row and 2 columns. So, its dimensions are $1 imes 2$.

The second matrix, $\left[\begin{array}{ll}1 & -4\end{array}\right]$, also has 1 row and 2 columns. Its dimensions are $1 imes 2$.

Now, let's check the compatibility rule. For the multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix.

• Number of columns in the first matrix = 2
• Number of rows in the second matrix = 1

Since 2 is not equal to 1, the number of columns in the first matrix does not match the number of rows in the second matrix. Therefore, Option A is not a possible matrix multiplication.

It's important to remember that even if the matrices have the same number of elements, their dimensions are what matter for multiplication compatibility. Here, both are $1 imes 2$ matrices, but their arrangement as rows and columns prevents them from being multiplied in this order.

Option B: $\left[\begin{array}{c}3 \\ -2\end{array}\right] \times\left[\begin{array}{cc}-1 & 0 \\ 0 & 1\end{array}\right]$

Let's determine the dimensions for Option B. The first matrix, $\left[\begin{array}{c}3 \\ -2\end{array}\right]$, has 2 rows and 1 column. Its dimensions are $2 imes 1$.

The second matrix, $\left[\begin{array}{cc}-1 & 0 \\ 0 & 1\end{array}\right]$, has 2 rows and 2 columns. Its dimensions are $2 imes 2$.

Now, let's apply the compatibility rule:

• Number of columns in the first matrix = 1
• Number of rows in the second matrix = 2

Since 1 is not equal to 2, the number of columns in the first matrix does not match the number of rows in the second matrix. Therefore, Option B is also not a possible matrix multiplication.

In this case, the first matrix is a column vector, and the second is a $2 imes 2$ matrix. For their product to be defined, the column vector would need to have a number of elements equal to the number of rows in the second matrix (which is 2). Since it only has 1 column, the multiplication is not compatible.

It's easy to get confused, especially when dealing with vectors, which are essentially matrices with only one row or one column. Always double-check the dimensions carefully – the number of columns in the first matrix and the number of rows in the second matrix. If they don't match, you can't proceed.

The Crucial Role of Dimensions in Matrix Multiplication

Let's reiterate the importance of dimensions. When we consider matrix multiplication, we are essentially looking at how rows of the first matrix interact with columns of the second. If the number of entries in a row of the first matrix doesn't match the number of entries in a column of the second matrix, the dot product operation that forms the elements of the resulting matrix cannot be performed. This is the fundamental reason behind the compatibility rule. It's not an arbitrary rule; it's derived from the very definition of how matrix multiplication is constructed.

Consider a general matrix $A$ of size $m imes n$ and a general matrix $B$ of size $p imes q$. For the product $AB$ to be defined, we must have $n = p$. If this condition is met, the resulting matrix $C = AB$ will have the dimensions $m imes q$. The inner dimensions ($n$ and $p$) must be equal, and the outer dimensions ($m$ and $q$) define the size of the product.

In Option A, we had a $1 imes 2$ matrix multiplied by a $1 imes 2$ matrix. The inner dimensions were 2 and 1. Since $2 \neq 1$, multiplication is impossible. The result would have been a $1 imes 2$ matrix if it were possible.

In Option B, we had a $2 imes 1$ matrix multiplied by a $2 imes 2$ matrix. The inner dimensions were 1 and 2. Since $1 \neq 2$, multiplication is impossible. The result would have been a $2 imes 2$ matrix if it were possible.

This concept is often a stumbling block for students learning linear algebra. They might look at the numbers within the matrices or the total number of elements and try to find a pattern that isn't there. The key is to always write down the dimensions of each matrix first and then compare the number of columns of the first with the number of rows of the second. This systematic approach prevents confusion and ensures accuracy.

Furthermore, matrix multiplication is not commutative, meaning that even if $A imes B$ is possible, $B imes A$ might not be, or if both are possible, they might yield different results. For $B imes A$ to be possible, the number of columns in $B$ must equal the number of rows in $A$. So, if $A$ is $m imes n$ and $B$ is $n imes q$, then $AB$ is $m imes q$. For $BA$, $B$ is $n imes q$ and $A$ is $m imes n$. The compatibility condition is $q = m$. If this holds, $BA$ is $n imes n$.

Understanding the dimensions is paramount. It's the gatekeeper to performing valid matrix operations. Without this understanding, any attempts to multiply matrices will likely lead to errors, whether in manual calculations or in programming applications that use matrix libraries.

Conclusion: Which Matrix Multiplication is Possible?

After carefully analyzing both options using the fundamental rule of matrix multiplication (the number of columns in the first matrix must equal the number of rows in the second matrix), we can definitively conclude:

• Option A: $\left[\begin{array}{ll}0 & 3\end{array}\right]_{1 \times 2} \times \left[\begin{array}{ll}1 & -4\end{array}\right]_{1 \times 2}$ is not possible because $2 \neq 1$.
• Option B: $\left[\begin{array}{c}3 \\ -2\end{array}\right]_{2 \times 1} \times \left[\begin{array}{cc}-1 & 0 \\ 0 & 1\end{array}\right]_{2 \times 2}$ is not possible because $1 \neq 2$.

Therefore, based on the provided options, neither matrix multiplication is possible.

Remember, the order matters, and the dimensions are key. Always check: (Rows of A x Columns of A) x (Rows of B x Columns of B). For A x B to be valid, Columns of A must equal Rows of B.

For further exploration into the fascinating world of matrices and linear algebra, you can consult resources like:

• Khan Academy's Linear Algebra section offers comprehensive explanations and practice problems.
• Wikipedia's entry on Matrix Multiplication provides a detailed mathematical overview.