Problem 17 - Precalculus: Functions and Graphs

Question

Find, if possible, \(A B\) and \(B A\). $$A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right], \quad B=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

Both \(AB\) and \(BA\) equal matrix \(A\).

Step 1: Check Matrix Dimensions

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \(A\) is a 3x3 matrix, and matrix \(B\) is also a 3x3 matrix. Therefore, both \(AB\) and \(BA\) are possible and both will result in 3x3 matrices.

Step 2: Calculate \(AB\)

Multiply matrix \(A\) by matrix \(B\) using the standard matrix multiplication method: each element in resultant matrix \(AB_{ij}\) is calculated as: \(AB_{ij} = \sum_k A_{ik} B_{kj}\). Since \(B\) is the identity matrix, \(AB\) will be the same as \(A\), which is: \[AB = \left[\begin{array}{lll}1 & 2 & 3 \4 & 5 & 6 \7 & 8 & 9\end{array}\right]\]

Step 3: Calculate \(BA\)

Multiply matrix \(B\) by matrix \(A\) using the same method as above. Matrix \(B\) is the identity matrix, so the resultant matrix \(BA\) will be the same as \(A\), thus: \[BA = \left[\begin{array}{lll}1 & 2 & 3 \4 & 5 & 6 \7 & 8 & 9\end{array}\right]\]

Step 4: Conclude the Multiplication

Both \(AB\) and \(BA\) result in matrix \(A\), as matrix \(B\) is the identity matrix. The identity matrix does not change the other matrix in multiplication.

Key Concepts

Identity Matrix

The identity matrix is a special type of matrix that is very important in linear algebra. It acts as the neutral element in matrix multiplication, just like the number 1 does in multiplication of real numbers. The identity matrix is always a square matrix, meaning it has the same number of rows and columns. In a 3x3 identity matrix, this would look like:

• Ones on the diagonal from the top left to the bottom right.
• All other elements are zeros.

For example, the matrix\[I_3 = \left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]\]is a 3x3 identity matrix. When you multiply any matrix by an identity matrix of compatible size, the original matrix is unchanged. It's like saying: anything times one is still itself. This is why in the exercise, multiplying either \(A\) by \(B\) or \(B\) by \(A\) results in \(A\) itself.

Matrix Dimensions

Matrix dimensions refer to the size of the matrix, typically given as \(m\times n\) where \(m\) is the number of rows and \(n\) is the number of columns. Understanding matrix dimensions is crucial for performing matrix multiplications.

In the exercise, both matrices \(A\) and \(B\) are described as 3x3 matrices. This means both matrices have 3 rows and 3 columns.
When multiplying two matrices, you need to ensure that the number of columns in the first matrix matches the number of rows in the second one. Therefore, only matrices of matching dimensions can be multiplied directly.
For instance, a 3x3 matrix can only multiply another 3x3 matrix, or equivalently, a matrix that has 3 rows. Thus, calculating \(AB\) and \(BA\) is perfectly feasible because both \(A\) and \(B\) satisfy these dimension requirements.

Matrix Properties

Matrices have several properties that define how they behave, especially during multiplication.
Understanding these properties helps in predicting the results of many operations.

• Associative Property: This means that \((AB)C = A(BC)\). You can change the grouping of matrices without changing the result.
• Distributive Property: The distributive property over matrix addition is that \(A(B + C) = AB + AC\), similar to basic arithmetic.
• Commutative Property: Unlike the properties above, most matrices do not exhibit commutative behavior, which means \(AB eq BA\) in general. However, in our exercise, because matrix \(B\) is an identity matrix, \(AB = BA = A\). This unique result showcases how the identity matrix stood as an exception.

3x3 Matrix

A 3x3 matrix is a type of square matrix with three rows and three columns. In various fields like computer graphics, physics, and engineering, 3x3 matrices help to perform complex calculations with ease.
A 3x3 matrix can be represented as follows:\[A = \left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{array}\right]\]Each element \(a_{ij}\) corresponds to the entry in the \(i^{th}\) row and \(j^{th}\) column.
There are special operations you can perform with 3x3 matrices such as inverses, determinants, and solving systems of linear equations. Understanding how to manipulate such datasets is key in linear algebra.
In the exercise, both matrix \(A\) and matrix \(B\) are 3x3 matrices, allowing them to be multiplied together successfully, resulting in outcomes that uphold the properties and dimensions described earlier.