Problem 158 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find \(a, b, c, d\) if \(\left[\begin{array}{ll}a+3 & 2 b-8 \\ c+1 & 4 d-6\end{array}\right]=\left[\begin{array}{cc}0 & -6 \\ -3 & 2 d\end{array}\right]\)

Step-by-Step Solution

Answer

So, the values of a, b, c, and d are -3, 1, -4, and 3 respectively.

Step 1: Formulate Equations

Begin by setting corresponding elements from each matrix equal to each other to form equations. This will result in: (1) \(a + 3 = 0\), (2) \(2b - 8 = -6\), (3) \(c + 1 = -3\), and (4) \(4d - 6 = 2d\).

Step 2: Solve for a

Solve the first equation \(a + 3 = 0\) for \(a\). Subtract 3 on both sides which gives: \(a = -3\).

Step 3: Solve for b

Solve the second equation \(2b - 8 = -6\) for \(b\). First, add 8 to both sides: \(2b = 2\). Then divide by 2 on both sides: \(b = 1\).

Step 4: Solve for c

Solve the third equation \(c + 1 = -3\) for \(c\). Subtract 1 from both sides which gives: \(c = -4\).

Step 5: Solve for d

Solve the fourth equation \(4d - 6 = 2d\) for \(d\). Subtract 2d on both sides: \(2d -6 = 0\). Then add 6 to both sides: \(2d = 6\). Divide by 2 on both sides gives: \(d = 3\)

Key Concepts

Matrix Equality

Matrix equality is a fundamental concept in linear algebra, and understanding it is essential for solving problems involving matrices. Two matrices are said to be equal if they have the same dimensions and their corresponding elements are identical.
In this exercise, we have two matrices that need to be equal, so we equate their corresponding elements:

• The element at the first row, first column in both matrices is equated as: \( a + 3 = 0 \).
• The element at the first row, second column is set as: \( 2b - 8 = -6 \).
• The second row, first column gives us: \( c + 1 = -3 \).
• Finally, the second row, second column results in: \( 4d - 6 = 2d \).

Recognizing matrix equality enables us to create and solve these linear equations efficiently. It's pivotal to verify each position's equivalence for solving matrix-related problems like this.

Linear Equations

Linear equations are equations of the first order, and they are vital in solving for unknown variables. In our given matrices, once the matrix equality has been established, each correspondence results in a linear equation.
Solve these equations by isolating the variable:

• From \( a + 3 = 0 \), subtract 3 to find \( a = -3 \).
• In \( 2b - 8 = -6 \), start by adding 8 to get \( 2b = 2 \), then divide by 2, hence \( b = 1 \).
• For \( c + 1 = -3 \), subtract 1 to derive \( c = -4 \).
• With \( 4d - 6 = 2d \), subtract \( 2d \) and add 6, leading to \( 2d = 6 \). Divide by 2 to find \( d = 3 \).

Efficiently solving linear equations is a key skill that forms the backbone of working with matrix equations, enabling students to determine unknowns.

Matrix Algebra

Matrix algebra encompasses operations related to matrices including addition, subtraction, and multiplication. In this exercise, solving a matrix equation requires understanding how variables within matrices interact.
Though this exercise primarily illustrates matrix equality, it's influential in preparing students for more complex operations in matrix algebra.

• Understand that in matrix addition or subtraction, you perform operations element-wise.
• Multiplication, which isn't needed here, requires a different approach involving summations of products.

By mastering the basics of matrix finding equations, students set the stage for tackling more sophisticated tasks such as solving systems of linear equations using matrices or working with matrix transformations. Understanding the order and operations in matrix algebra ensures accuracy and proficiency when approaching matrix problems.