Problem 176 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

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

Step-by-Step Solution

Answer

The product of matrices \(A\) and \(B\) is \(AB=\left[\begin{array}{ll}0 & 0 \ 0 & 0\end{array}\right]\).

Step 1: Multiply First Row of Matrix A with First Column of Matrix B

Firstly, \(A[1,1]*B[1,1]+A[1,2]*B[2,1]=1*0+0*0=0\) is the entry in first row and first column of the resulting matrix.

Step 2: Multiply First Row of Matrix A with Second Column of Matrix B

Now, \(A[1,1]*B[1,2]+A[1,2]*B[2,2]=1*0+0*1=0\) gives the entry in the first row and second column of the resulting matrix.

Step 3: Multiply Second Row of Matrix A with First Column of Matrix B

Following, \(A[2,1]*B[1,1]+A[2,2]*B[2,1]=0*0+0*0=0\) gives the entry in the second row and first column of the resulting matrix.

Step 4: Multiply Second Row of Matrix A with Second Column of Matrix B

Finally, \(A[2,1]*B[1,2]+A[2,2]*B[2,2]=0*0+0*1=0\) is the entry in the second row and second column of the resulting matrix.

Step 5: Form the Resultant Matrix

Combine all the calculated values to form the resultant matrix.

Key Concepts

Matrix Theory

Matrix theory is a branch of mathematics that focuses on the study of matrices and their properties. A matrix is essentially a rectangular array of numbers or functions arranged in rows and columns. The fundamental importance of matrices lies in their ability to represent linear transformations and systems of linear equations.
Mathematically, matrices are used to:

• Describe complex systems or structures, such as networks or economic models.
• Perform linear transformations in vector spaces.
• Facilitate operations like addition, multiplication, and scalar products.

In our exercise, we are working with two specific matrices: \(A\) and \(B\). Understanding the characteristics of matrices, like dimensions (number of rows by columns), is crucial in determining how matrices can interact with each other, particularly for operations like multiplication. It's also important to note that matrix dimensions must align correctly for multiplication to be possible; the number of columns in the first matrix must match the number of rows in the second matrix.

Matrix Operations

Matrix operations involve executing mathematical calculations on matrices, similar to calculations on numbers. However, matrices come with their own unique set of rules.
Matrix multiplication, in particular, isn't as straightforward as element-wise multiplication. It involves a methodical process where the elements of rows from the first matrix are multiplied by the corresponding elements of the columns of the second matrix, and then added together to produce a new matrix.
In our exercise, to find \(A \times B\), each element of the resulting matrix is obtained by taking the dot product of rows from \(A\) with columns from \(B\). For instance:

• The entry in the first row and first column is derived by multiplying the first row of \(A\) with the first column of \(B\) and adding the results: \[ A[1,1]*B[1,1] + A[1,2]*B[2,1] = 1\times0 + 0\times0 = 0 \]
• This process is repeated for each combination of rows and columns, resulting in a complete new matrix.

Being systematic is key to calculating the product of two matrices correctly.

Linear Algebra

Linear algebra is the mathematical discipline that deals with vectors, vector spaces, linear transformations, and systems of linear equations. Matrices are a central topic within linear algebra as they provide a compact way to handle linear equations and transformations.
When dealing with matrices, it’s essential to recognize the principles of linearity, concerning how matrix operations maintain the properties of linear transformations. Multiplication of matrices reflects these principles by transforming one vector space into another, maintaining the structure of the original form.

• In the context of our exercise, multiplying matrices \(A\) and \(B\) provides a new matrix that represents transformations encoded by those matrices.
• Understanding how these transformations work helps in applications ranging from computer graphics, where transformations are critical, to differential equations and quantum mechanics.

Linear algebra also assists in solving systems of equations by representing them as matrix equations, offering efficient solutions through computation, which is significant in applied mathematics and computer science.