Numerical elements MCQ Quiz in తెలుగు - Objective Question with Answer for Numerical elements - ముఫ్త్ [PDF] డౌన్‌లోడ్ కరెన్

Concept:

Properties of Determinant of a Matrix:

• If each entry in any row or column of a determinant is 0, then the value of the determinant is zero.
• For any square matrix say A, |A| = |AT|.
• If we interchange any two rows (columns) of a matrix, the determinant is multiplied by -1.
• If any two rows (columns) of a matrix are same then the value of the determinant is zero.

Calculation:

\(\begin{vmatrix} 2 & 7 & 37\\ 3& 6 & 33\\ 4 & 5 & 29 \end{vmatrix}\)

Apply C2 → 5C2 + C₁, we get

= \(\begin{vmatrix} 2 & 37 & 37\\ 3& 33 & 33\\ 4 & 29 & 29 \end{vmatrix}\)

As we can see that the second and the third column of the given matrix are equal. 

We know that, if any two rows (columns) of a matrix are same then the value of the determinant is zero.

∴ \(\begin{vmatrix} 2 & 7 & 37\\ 3& 6 & 33\\ 4 & 5 & 29 \end{vmatrix}\) = 0

\( P=\begin{bmatrix} 1 & \alpha & 3\ 1 & 3 & 3\ 2 & 4 & 4 \end{bmatrix} \)

For 3X3 matrix,

\( |Adj A|=|A|^{2} \)

\( |Adj A|=16 \)

\( 1(12-12)-\alpha(4-6)+3(4-6)=16 \)

\( 2\alpha-6=16 \)

\( 2\alpha=22 \)

\( \alpha=11 \)

Calculation

|p| = 2α - 6 = (det A)² = 16

⇒ α = 11

Hence option 2 is correct

Concept:

The co-factor of A_ij = (– 1)^i+jM_ij, where Mij is the minor obtained by removing the i^th row and j^th column.

Calculation:

Given, A = \(\rm \begin{bmatrix}\beta&\alpha&3\\ \alpha&\alpha&\beta\\\ -\beta &\alpha&2\alpha\end{bmatrix}\) and B = \(\rm\begin{bmatrix}3\alpha&-9&3\alpha\\\ -\alpha&7&-2\alpha\\\ -2\alpha&5&-2\beta\end{bmatrix}\)

Now, equating co-factor fo A₂₁  = (2α² – 3α) = α

⇒ α = 2, 0

⇒ α = 2 [∵ α ≠ 0] 

Now, 2α² – αβ = 3α 

⇒ 8 – 2β = 6

⇒ β = 1

∴ |AB| = |A cof (A)| = |A|³ 

∴ |A| = \(\left|\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 2 & 1 \\ -1 & 2 & 4 \end{array}\right|\) = 6 - 2(9) + 3(6) = 6

⇒ |A|3 = 63 = 216

∴ The value of det(AB) is equal to 216.

The correct answer is Option 4.

Concept:

Determinants:

• A linear combination of the rows/columns does not affect the value of the determinant.
• If two rows/columns of a given matrix are interchanged, then the value of the determinant gets multiplied by - 1.
• If a row/column of a given matrix is multiplied by a scalar k, then the value of the determinant is also multiplied by k.

Calculation:

Let D = \(\begin{vmatrix}5^2&5^3&5^4\\5^3&5^4&5^5\\5^5&5^6&5^7\end{vmatrix}\).

Dividing R₁ by scalar 5² and R₂ by 5³, we get:

⇒ D = \((5^2)(5^3)\begin{vmatrix}1&5&5^2\\1&5&5^2\\5^5&5^6&5^7\end{vmatrix}\)

Using R₁ → R₁ - R₂, we get:

⇒ D = \((5^2)(5^3)\begin{vmatrix}0&0&0\\1&5&5^2\\5^5&5^6&5^7\end{vmatrix}\)

Expanding along R₁, we get:

⇒ D = 0.