Question:
If
\[
A=
\begin{bmatrix}
2026 & 2025 & 2024
2025 & 2024 & 2023
2024 & 2023 & 2022
\end{bmatrix},
\]
then \(\left|A^{2026}-A^{2025}\right|=\)
If
\[
A=
\begin{bmatrix}
2026 & 2025 & 2024
2025 & 2024 & 2023
2024 & 2023 & 2022
\end{bmatrix},
\]
then \(\left|A^{2026}-A^{2025}\right|=\)
Show Hint
If any matrix factor has determinant zero, then the determinant of the product is also zero.
Updated On: Jul 18, 2026
- \(2026\)
- \(2025\)
- \(2024\)
- \(0\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Observe that& nbsp;
\[ A^{2026}-A^{2025} = A^{2025}(A-I). \]
Taking determinants,
\[ \left|A^{2026}-A^{2025}\right| = |A^{2025}|\;|A-I|. \]
Now,
\[ A-I= \begin{bmatrix} 2025 & amp; 2025 & amp; 2024\\ 2025 & amp; 2023 & amp; 2023\\ 2024 & amp; 2023 & amp; 2021 \end{bmatrix}. \]
Since
\[ R_1-2R_2+R_3=0, \]
the rows are linearly dependent.
Hence,
\[ |A-I|=0. \]
Therefore,
\[ \boxed{\left|A^{2026}-A^{2025}\right|=0.} \]
Thus, the correct option is
\[ \boxed{(D)}. \]
Was this answer helpful?
0
0
Top TS PGECET Mathematics Questions
- If \[ \{(2,-3,5),\ (1,\alpha,7),\ (3,\beta,3)\} \] is not a basis of the vector space \(\mathbb{R}^3\), then \((\alpha,\beta)\) lies on the locus:
Consider the linear system of equations \[ \begin{bmatrix} 3 & -1 & 4 \\ 6 & 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}. \] In this system of equations, if \(x\) is always a fixed constant, then the system has:& nbsp;
- Evaluate \[ \iint_D x^2y\,dx\,dy, \] where \[ D:\;x^2+y^2\le16. \]
- If \[ \overline{V}=x^{2}y\,\overline{i}-yz^{2}\,\overline{j}+xyz\,\overline{k} \] is the linear velocity of a particle in circular motion at the point \((1,1,1)\), then its angular velocity is
- Evaluate \[ \oint_C \frac{z+2}{z^2+2z+2}\,dz, \] where \(C\) is the circle \[ |z-\alpha|=1, \qquad \alpha=-1+\frac{3}{2}i. \]
View More Questions
Top TS PGECET Matrices and Determinants Questions
Top TS PGECET Questions
- If \( P = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix} \) is the modal matrix of \( A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{pmatrix} \) then the sum of all the elements of \( P^{-1}AP \) is
- If the system of equations \( kx + y + z = k - 1 \), \( x + ky + z = k - 1 \), and \( x + y + kz = k - 1 \) has infinite solutions, then \( k \) is
- The value of the integral \( \int_C (2xy - x^2) \, dx + (x^2 + y^2) \, dy \) where \( C \) is the boundary of the region enclosed by \( y = x^2 \) and \( y^2 = x \), described in the positive sense, is
- Let \( f(x) = \log x \). The number \( C \) strictly between \( e^2 \) and \( e^3 \) such that its reciprocal is equal to \( \frac{f(e^3) - f(e^2)}{e^3 - e^2} \) is
- The solution of the differential equation \( (D^2 + 2)y = x^2 \) is
View More Questions