举一反三
- Ais an[img=44x19]1803b6a4c8dddc0.png[/img]matrix. A is not invertible if and only if 0 is an eigenvalue of A.
- The matrix A is[img=44x19]1803b6ad8347a80.png[/img]. If the eqaution Ax=0 has only the trivial solution , thenA is invertible.
- Which of the following statement is false? A: If[img=77x23]1803b6aa763b00a.png[/img], where[img=77x23]1803b6aa7e8aab9.png[/img] and D is a diagonal matrix, then B is a symmetric matrix. B: An orthogonal matrix is orthogonally diagonalizable. C: An[img=44x19]1803b6aa8680eb5.png[/img]symmetric matrix has n real eigenvalues. D: An[img=44x19]1803b6aa8680eb5.png[/img]symmetric matrix hasnlinearly independent eigenvectors.
- Which of the following statement is false? A: Ais an[img=49x19]1803b6abf33899a.png[/img]matrix.[img=66x23]1803b6abfb9f77f.png[/img]is a QR factorization ofA. The columns ofQform an orthonormal basis for the column space ofA. B: If A=QR, where Q has orthonormal columns, then[img=79x27]1803b6ac036f846.png[/img] C: Ahas a QR factorization, then A must be a square matrix. D: Any matrix can have a QR factorization. E: If a matrix A has a QR factorization, then the ranks of A and Q are the same.
- The columns of the standard matrix for a linear transformation from [img=22x19]1803b6a8c19a71a.png[/img]to [img=25x19]1803b6a8c9f3863.png[/img] are the images of the columns of the [img=44x19]1803b6a8d3fc1b1.png[/img] indentity matrix.
内容
- 0
A is an [img=36x19]1803c4e61381e6b.png[/img] matrix and [img=12x14]1803c4e61b3a2fb.png[/img] is an n-dimensional column vector. If the [img=201x51]1803c4e628b33fd.png[/img], then the linear system ____________ A: [img=60x19]1803c4e6316cf85.png[/img] must have infinitly many solutions. B: [img=139x51]1803c4e63cf2009.png[/img]must have nontrivial solutions. C: [img=60x19]1803c4e6316cf85.png[/img] must have a unique solution. D: [img=139x51]1803c4e63cf2009.png[/img] only has nontrivial solutions.
- 1
Let A be a [img=41x19]1803c4e38325ee9.png[/img] matrix, then ________. A: If Ax=0 has only trivial solution, then Ax=b must have a unique solution. B: If Ax=0 has a nontrivial solution, then Ax=b must have infinitely many solutions. C: If Ax=b has a unique solution, then Ax=0 must have a unique solution. D: If Ax=0 has a nontrivial solution, then [img=55x23]1803c4e38c0c9df.png[/img] must have a nontrivial solution.
- 2
Let [img=99x19]1803c4e7d6c70bb.png[/img] be a linear transformation. If A is the standard matrix representation of [img=13x19]1803c4e7de70caa.png[/img], then an [img=44x19]1803c4e7e5f3add.png[/img] matrix B will also be a matrix representation of L if and only if B is similar to A.
- 3
An[img=49x19]1803b6aafbee947.png[/img] matrix A has orthonormal columns if and only if[img=59x23]1803b6ab03fb852.png[/img].
- 4
Let A be a 4×4 matrix, the determinant of A is 1/3, and [img=21x20]1803c4e4937ebc3.png[/img] the adjoint matrix of A, then [img=115x27]1803c4e49b725a2.png[/img]=________. A: 1 B: 3 C: 6 D: 9