Determinant Change Factor for Row Multiplication
To solve this problem, we need to understand a fundamental property of determinants concerning scalar multiplication of a matrix row.
Determinant Property
- If a single row (or column) of a matrix is multiplied by a scalar value, say '$k$', then the determinant of the resulting matrix is '$k$' times the determinant of the original matrix.
- Mathematically, if $A$ is a matrix and $A'$ is the matrix obtained by multiplying one row of $A$ by $k$, then $\det(A') = k \times \det(A)$.
Applying the Property
In this specific question:
- The matrix is a $3 \times 3$ matrix.
- All elements of one row are multiplied by the scalar $k=3$.
Using the property stated above:
The determinant of the new matrix will be 3 times the determinant of the original matrix.
Therefore, the determinant changes by a factor of 3.
Important Questions from Determinants
- An $n \times n$ square matrix $A$ satisfies $A^T = A^{-1}$. The determinant of this matrix may take which of the following value(s)?
The determinant of the matrix $\begin{pmatrix} 1 & p & q + r \\ 1 & q & r + p \\ 1 & r & p + q \end{pmatrix}$ is ______.
Let $f(x) = \begin{vmatrix} x^3 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^2 & p^3 \end{vmatrix}$
where $p$ is a constant.
The value of $\frac{d^3}{dx^3}f(x)$ at $x = 0$ is
How many matrices of different orders are possible with elements comprising all prime numbers less than 30?
If A is a square matrix of order 3 and det A = 5, then what is det [(2A) -1 ] equal to ?
