Determinant Calculation for Scaled 4x4 Matrix
We are given a $4 \times 4$ matrix $A$ such that its determinant is 3.
- Given: $\det(A) = 3$
- Matrix size: $n=4$
We need to find the determinant of the matrix $2A$.
Determinant Property Application
To find the determinant of $2A$, we use the property of determinants concerning scalar multiplication:
$ \det(kA) = k^n \det(A) $
where $k$ is a scalar, $A$ is an $n \times n$ matrix, and $n$ is the dimension of the matrix.
Calculating Determinant of 2A
In this case, $k=2$ and $n=4$. Applying the property:
- Substitute the values into the formula:
$ \det(2A) = 2^4 \times \det(A) $
- Substitute the given determinant value, $\det(A) = 3$:
$ \det(2A) = 2^4 \times 3 $
- Calculate $2^4$:
$2^4 = 16$
- Perform the final multiplication:
$ \det(2A) = 16 \times 3 = 48 $
The value of the determinant of $2A$ is 48.
Important Questions from Determinants
If \(A=\left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 3 & 0 \\ 1 & 0 & 1 \end{array}\right]\), then what is the value of det[adj(adjA)] ?
If A, B and C are square matrices of order 3 and det(BC) = 2 det(A), then what is the value of det(2A-1BC)?
If \(A=\left[\begin{array}{rrr} 0 & 3 & 4 \\ -3 & 0 & 5 \\ -4 & -5 & 0 \end{array}\right]\), then which one of the following statements is correct?
If \(\left|\begin{array}{ccc} x^2+3 x & x-1 & x+3 \\ x+1 & -2 x & x-4 \\ x-3 & x+4 & 3 x \end{array}\right|\) = ax4 + bx3 + cx2 + dx + e, then what is the value of e?"
If all elements of a third order determinant are equal to 1 or -1, then the value of the determinant is:
