Numerical elements MCQ Quiz in मराठी - Objective Question with Answer for Numerical elements - मोफत PDF डाउनलोड करा

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:

$\left|\begin{array}{ccc}2& 7& 37\\ 3& 6& 33\\ 4& 5& 29\end{array}\right|$

Apply C2 → 5C2 + C₁, we get

= $\left|\begin{array}{ccc}2& 37& 37\\ 3& 33& 33\\ 4& 29& 29\end{array}\right|$

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.

∴ $\left|\begin{array}{ccc}2& 7& 37\\ 3& 6& 33\\ 4& 5& 29\end{array}\right|$ = 0

$P=\left[\begin{array}{ccccccc}1& \alpha & 31& 3& 32& 4& 4\end{array}\right]$

For 3X3 matrix,

$|AdjA|=|A{|}^2$

$|AdjA|=16$

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

$2\alpha -6=16$

$2\alpha =22$

$\alpha =11$

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 = $\left[\begin{array}{ccc}\beta & \alpha & 3\\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{array}\right]$ and B = $\left[\begin{array}{ccc}3\alpha & -9& 3\alpha \\ -\alpha & 7& -2\alpha \\ -2\alpha & 5& -2\beta \end{array}\right]$

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 = $\left|\begin{array}{ccc}{5}^2& 5^3& 5^4\\ 5^3& 5^4& 5^5\\ 5^5& 5^6& 5^7\end{array}\right|$.

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

⇒ D = $(5^2)(5^3)\left|\begin{array}{ccc}1& 5& 5^2\\ 1& 5& 5^2\\ 5^5& 5^6& 5^7\end{array}\right|$

Using R₁ → R₁ - R₂, we get:

⇒ D = $(5^2)(5^3)\left|\begin{array}{ccc}0& 0& 0\\ 1& 5& 5^2\\ 5^5& 5^6& 5^7\end{array}\right|$

Expanding along R₁, we get:

⇒ D = 0.