Let A denote the matrix A = bmatrix 2 & 0 & 1 1 & 1 & 0 1 & 0 & 1… - Solution | JEE Mathematics

MCQMediumJEE 2024Solving Linear Equations (Matrix Method)

Let $A$ denote the matrix $A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$ , and $B = [B_1, B_2, B_3]$ where $B_1$ , $B_2$ , and $B_3$ are column matrices such that:

AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}, \quad AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}

If $\alpha = |B|$ and $\beta$ is the sum of all diagonal elements of $B$ , then $\alpha^3 + \beta^3$ is equal to:

A
$28$
B
$29$
C
$30$
D
$31$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given:

A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}, \quad B = [B_1, B_2, B_3]

with

AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}, \quad AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}

Find: $\alpha^3 + \beta^3$ where $\alpha = |B|$ and $\beta$ is the sum of diagonal elements of $B$ .

From the solution, let

B_1 = \begin{bmatrix} x_1 \\ y_1 \\ z_1 \end{bmatrix}, \quad B_2 = \begin{bmatrix} x_2 \\ y_2 \\ z_2 \end{bmatrix}, \quad B_3 = \begin{bmatrix} x_3 \\ y_3 \\ z_3 \end{bmatrix}

Working from the Given Systems

Using matrix multiplication, the solution gives the systems: For $B_1$ ,

\begin{cases} 2x_1 + z_1 = 1 \\ x_1 + y_1 = 0 \\ x_1 + z_1 = 0 \end{cases}

For $B_2$ ,

\begin{cases} 2x_2 + z_2 = 2 \\ x_2 + y_2 = 3 \\ x_2 + z_2 = 0 \end{cases}

For $B_3$ ,

\begin{cases} 2x_3 + z_3 = 3 \\ x_3 + y_3 = 2 \\ x_3 + z_3 = 1 \end{cases}

Answer from the Extracted Conclusion

The solution concludes:

\alpha = |B| = 3, \quad \beta = 1

Hence,

\alpha^3 + \beta^3 = 27 + 1 = 28

Therefore, the correct option is A.

The solution contains a formatting inconsistency in Step 4, but its final concluded values and final result clearly give $28$ .

Common mistakes

Treating $B_1, B_2, B_3$ as row matrices is incorrect because the question explicitly states they are column matrices. Use column-wise multiplication in $AB_i$ .
Assuming $B$ is given directly is wrong. You must infer the columns of $B$ from the three relations $AB_1$ , $AB_2$ , and $AB_3$ .
Confusing $\beta$ with the sum of all entries of $B$ is incorrect. $\beta$ is the sum of diagonal elements, that is, the trace of $B$ .

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Answer

Step-by-step solution

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