If X = [x, y, z]ᵀ is a solution of the system of equations AX = B, where adj A = [[4, 2, 2], [-5, 0,

Q: Wait, can't I just find $A$ by taking the adjoint of the adjoint matrix? Why is that wrong?

That is a classic trap! While it is true that $\text{adj}(\text{adj } A) = |A|^{n-2} A$, it is computationally expensive and unnecessary. We only need $X$, not $A$ itself. Always look for the most direct path to the target variable!

Q: Why do we get $\pm 10$ for the determinant? Shouldn't it be just $10$?

Excellent observation! When we solve $|A|^2 = 100$, we are solving a quadratic equation. Mathematically, both $10$ and $-10$ satisfy the equation. In JEE Advanced, ignoring the negative root is a common way to lose marks. Always carry the $\pm$ until the final absolute value operation!

Q: Do I have to solve for $x, y,$ and $z$ individually? That seems like a lot of work.

You have a sharp eye for efficiency! You do not need to find $x, y,$ and $z$ separately. Notice that the question asks for the sum $x+y+z$. You can sum the components of the resulting vector directly after the matrix multiplication. Work smarter, not harder!

Question

If $X = \begin{bmatrix} x \ y \ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $	ext{adj } A = \begin{bmatrix} 4 & 2 & 2 \ -5 & 0 & 5 \ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \ 0 \ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :

Options:

(A) 2

(B) $\frac{3}{2}$

(C) 1

(D) 3

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given a system of linear equations $AX = B$. Our objective is to determine the value of $|x + y + z|$ using the provided adjoint matrix $	ext{adj } A$ and the constant matrix $B$.

The common pitfall is attempting to calculate the matrix $A$ directly. Instead, we utilize the fundamental relationship:
$$X = A^{-1}B$$

Recalling the definition of the inverse matrix, $A^{-1} = \frac{1}{|A|} 	ext{adj } A$, we can rewrite the system as:
$$X = \frac{1}{|A|} (	ext{adj } A) B$$

This elegant substitution confirms that we do not need to find $A$ itself. We only require the determinant $|A|$ and the product of the adjoint matrix with $B$.

### The Determinant Mystery

To find $|A|$, we employ the standard property of determinants:
$$|	ext{adj } A| = |A|^{n-1}$$

Given that our matrix is $3 	imes 3$, we have $n=3$, which simplifies the relation to $|	ext{adj } A| = |A|^2$. Calculating the determinant of the provided adjoint matrix:
$$|	ext{adj } A| = 4(0 - (-10)) - 2(-15 - 5) + 2(10 - 0) = 40 + 40 + 20 = 100$$

Since $|A|^2 = 100$, we must account for both possible roots:
$$|A| = \pm 10$$

### The Final Synthesis

We calculate the product of the adjoint matrix and the constant matrix $B$:
$$(	ext{adj } A) B = \begin{bmatrix} 20 \ -10 \ 10 \end{bmatrix}$$

Substituting this into our master equation, we obtain:
$$X = \frac{1}{\pm 10} \begin{bmatrix} 20 \ -10 \ 10 \end{bmatrix}$$

This yields the individual variables:
$$x = \frac{20}{\pm 10}, \quad y = \frac{-10}{\pm 10}, \quad z = \frac{10}{\pm 10}$$

To find $|x + y + z|$, we sum the components first:
$$x + y + z = \frac{1}{\pm 10} (20 - 10 + 10) = \frac{20}{\pm 10} = \pm 2$$

Taking the absolute value of the result:
$$|\pm 2| = 2$$

The final answer is **2**.

Animated Solution for Mathematics - Matrices and Determinants: If $X = x y z$ is a solution of the system of equations $A X = B$ , where $adj A = 4 - 5 1 20 - 2 253$ and $B = 402$ , then $∣ x + y + z ∣$ is equal to :

Select Answer:

Visualized Solution

System of Equations $A X = B$

The Master Formula for $X$

Property of Determinants

Calculating $∣ adj A ∣$

Finding the value of $∣ A ∣$

Matrix Multiplication: $(adj A) \cdot B$

Expressing Matrix $X$

Summing $x + y + z$

Final Answer: $∣ x + y + z ∣$

The Sigma Insight: Adjoint and Inverse of a Matrix

Analyzing the Setup

The Determinant Mystery

The Final Synthesis

Similar Questions

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Comprehension Passage

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Visualized Solution

System of Equations .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } AX=B

The Master Formula for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X

Property of Determinants

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣adj A∣

Finding the value of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣A∣

Matrix Multiplication: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (adj A)⋅B

Expressing Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X

Summing .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x+y+z

Final Answer: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣x+y+z∣

The Sigma Insight: Adjoint and Inverse of a Matrix

Analyzing the Setup

The Determinant Mystery

The Final Synthesis

Similar Questions

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Comprehension Passage

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