Problem 187 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find \(x\) so that \(\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{lll}1 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 3 & 2\end{array}\right]\left[\begin{array}{l}1 \\ 1 \\ x\end{array}\right]=O\).

Step-by-Step Solution

Answer

Solving the quadratic equation, \(2x^2 + 10x + 17 = 0\), will provide the value for 'x'.

Step 1: Setting Up the Multiplication

Begin with the multiplication of the row vector and the square matrix. Carry out the matrix multiplication manually or with the help of a calculator. The resulting product is \(\left[1, 5x + 16, 2x + 5\right]\)

Step 2: Determine the Final Product

Multiply the resulting row vector from step 1 with the column vector. This results in a scalar. The process for this is multiplying each corresponding values in the two vectors together and then summing the products. The equation thus obtained is as follows: \(1 + (5x +16) + (2x^2 + 5x)\)

Step 3: Simplify and Solve for x

This resulting equation must be equal to zero as stated in the question. Simplify the equation and solve for 'x'. We obtain the equation as follows: \(1 + 16 + 5x + 5x + 2x^2 = 0\). This simplifies to \(2x^2 + 10x + 17 = 0\). Solving this quadratic equation will yield the value for 'x'.

Key Concepts

Understanding the System of Linear Equations

When we encounter a system of linear equations, we're looking at a collection of two or more linear equations with the same set of variables. These equations are called 'linear' because they represent straight lines when graphed on a coordinate plane. Systems of linear equations can have a single solution (where lines intersect at one point), no solution (where lines are parallel and never intersect), or infinitely many solutions (where lines overlap entirely).

In the context of matrix multiplication, like in our original exercise, we can use the concept of linear equations to understand how matrices interact. Each entry in a matrix can be considered as a coefficient of a variable in a linear equation. When we multiply matrices together, we're essentially combining multiple systems of linear equations into one. The result can give us a new system of linear equations or, as in the case of the exercise, a single scalar value that represents a condition (in this case, being equal to zero) that the system must satisfy.

Quadratic Equations Unveiled

Quadratic equations are fundamental in algebra and represent parabolas when graphed. Any equation of the form \( ax^2 + bx + c = 0 \) is considered a quadratic equation, where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The reason \( a \) cannot be zero is that, without the \( x^2 \) term, the equation would be linear.

The solutions to a quadratic equation are the 'x-intercepts' or 'roots' of the parabola and can be found using various methods, like factoring, completing the square, or the quadratic formula. In our exercise, after simplifying the product of matrices, we derived a quadratic equation \(2x^2 + 10x + 17 = 0\). Solving this will tell us the value of 'x' that satisfies the initial matrix equation.

Algebraic Operations and Matrix Multiplication

Algebraic operations are the building blocks of algebra, including addition, subtraction, multiplication, division, and the usage of exponents and radicals. Matrices obey specific rules for algebraic operations, and matrix multiplication is particularly unique. Unlike regular multiplication, matrix multiplication is not commutative (switching the order can yield different results), and it involves taking the dot product of rows and columns rather than element-wise multiplication.

The process we followed in the exercise showcases an algebraic operation involving matrices and a vector. Step 1 required us to multiply a row vector and a matrix, while Step 2 involved multiplying the resulting row vector with a column vector to get a scalar. The algebra rules applied in these steps are essential for understanding higher-level mathematics, computational algorithms, and many applications in physics and engineering.