Problem 177 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Given \(A=\left[\begin{array}{ll}1 & 0\end{array}\right]\) and \(B=\left[\begin{array}{l}0 \\ 1\end{array}\right]\), find \(A B\) and \(B A\).

Step-by-Step Solution

Answer

The multiplication \(A B\) results in a 1x1 matrix (or scalar) with the single element \(0\). The multiplication \(B A\) results in a 2x2 matrix with all elements \(0\).

Step 1: Calculate \(A B\)

Matrix \(A\) has dimensions 1x2 and \(B\) has dimensions 2x1. Hence, they can be multiplied and the resulting matrix will have dimensions 1x1. For the multiplication, the element of the resulting matrix is calculated as the dot product \(A B = (A_{11}\cdot B_{11}) + (A_{12}\cdot B_{21})\).

Step 2: Calculate \(B A\)

Here, matrix \(B\) has dimensions 2x1 and \(A\) has dimensions 1x2. Hence, they can be multiplied and the resulting matrix will have dimensions 2x2. Multiply each element of \(B\) by each element of \(A\) separately to fill the 2x2 matrix. The result would be \[ B A = \left[\begin{array}{ll}B_{11}\cdot A_{11} & B_{11}\cdot A_{12} \ B_{21}\cdot A_{11} & B_{21}\cdot A_{12} \end{array}\right]\].

Step 3: Plug in and simplify

Now we can substitute the actual numbers from matrices \(A\) and \(B\) into the multiplication definitions from steps 1 and 2, and simplify to obtain the results.

Key Concepts

Dot Product

The dot product is a fundamental concept in matrix multiplication. It's a way of multiplying vectors that results in a single number, known as a scalar. To calculate the dot product, you take corresponding elements from each of the two vectors, multiply them together, and then sum all these products.
For example, if you have two vectors, let's call them vector A: \((a_1, a_2, ..., a_n)\) and vector B: \((b_1, b_2, ..., b_n)\), the dot product, denoted as \(A \cdot B\), is calculated as:
\[ A \cdot B = a_1b_1 + a_2b_2 + ... + a_nb_n \]
In the context of our exercise, when calculating the product \(AB\) where \(A\) and \(B\) are matrices, we essentially apply the dot product to contributions from all suitable pairs of rows and columns.

• For the product \(AB\), the dot product is calculated by multiplying elements across rows of \(A\) by the corresponding columns of \(B\).
• This results in a single number, because \(A\) is a row vector and \(B\) is a column vector in this exercise.

Matrix Dimensions

Understanding matrix dimensions is crucial for performing operations like addition, subtraction, and especially multiplication. Matrix dimensions tell you the structure or shape of the matrix, specifying how many rows and columns it has. It's commonly denoted as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.
For matrix multiplication, the number of columns in the first matrix must match the number of rows in the second.

• In our example, matrix \(A\) has dimensions 1x2, meaning 1 row and 2 columns.
• Matrix \(B\) has dimensions 2x1, meaning 2 rows and 1 column.

This compatibility (2 columns in A and 2 rows in B) allows us to multiply these matrices.
Moreover, the resultant matrix will have dimensions based on the outer dimensions of the input matrices. Therefore, when multiplying \(AB\), you end up with a 1x1 matrix.
When you reverse this order and calculate \(BA\), since \(B\)'s rows match \(A\)'s columns (2x1 with 1x2), the resultant matrix has dimensions of 2x2.

Matrices A and B

In the exercise, we are given two specific matrices to work with: Matrix \(A\) and Matrix \(B\). Let's dig deeper into their roles and operations.
Matrix \(A\) is structured as: \(A = \left[ \begin{array}{ll} 1 & 0 \end{array} \right]\)- Matrix \(A\) is a 1x2 matrix, meaning it has 1 row and 2 columns.Matrix \(B\) is structured as:\(B = \left[ \begin{array}{l} 0 \ 1 \end{array} \right]\)- Matrix \(B\) is a 2x1 matrix, having 2 rows and 1 column.When calculating \(AB\), their dimensions align such that: - You perform operations element-wise across the row of \(A\) and the column of \(B\), using the dot product. - The result is a single value in a 1x1 matrix.Conversely, for \(BA\): - The multiplication takes each element in the two rows of \(B\) and multiplies them by both elements in the single row of \(A\). - This fills a 2x2 matrix, demonstrating how the dimensions and structure of original matrices influence the output.