若二阶实对称矩阵A与矩阵合同，则二次型X^TAX的标准形是________

A . 1 B . 3 C . 4 D . 5

设三阶实对称矩阵A的特征值为<img src='https://img2.soutiyun.com/ask/2020-12-21/977395452276495.png' />矩阵A的属于特征值<img src='https://img2.soutiyun.com/ask/2020-12-21/977395462822098.png' />的特征向量是<img src='https://img2.soutiyun.com/ask/2020-12-21/977395471307584.png' />试求矩阵A。

设f=x^TA x是一个实二次型， 若有实n维向量<img src='https://img2.soutiyun.com/ask/2020-12-29/978125347407201.png' />证明：必有实n维向量<img src='https://img2.soutiyun.com/ask/2020-12-29/97812535927688.png' />

已知二次型<img src='https://img2.soutiyun.com/ask/2020-12-29/978128441590178.png' />经过正交变换化为标准形<img src='https://img2.soutiyun.com/ask/2020-12-29/978128457965115.png' />求参数a、b及所用的正交变换矩阵。

证明:A是π阶方阵,对于任意<img src='https://img2.soutiyun.com/ask/2021-03-04/983705238463764.png' />有x^TAx=0的充分必要条件是A是反对称矩阵.

A.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787292961377.png' /> B.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787303034953.png' /> C.<img src='https://img2.soutiyun.com/ask/2020-08-20/96678731188946.png' /> D.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787321664019.png' />

如果实对称矩阵A与矩阵<img src='https://img2.soutiyun.com/ask/2020-08-20/966787469346466.png' />合同，则二次型x^TAx的规范形为(). A.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787498718146.png' /> B.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787513013964.png' /> C.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787525690689.png' /> D.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787536172288.png' />

A．A.a=2 B．B.a=1 C．C.a=0 D．D.a=-1

设A为实矩阵,B=AA^T,且<img src='https://img2.soutiyun.com/ask/2021-03-04/983706151007959.png' />则A=_______ .

A．AB-BA B．AB+BA C．(AB)² D．BAB

A．A.(2 4 0 0 5 − 8 0 0 5 ) B．B.(2 4 0 0 5 − 4 0 − 4 5 ) C．C.(2 2 0 2 5 − 4 0 − 4 5 ) D．D.(2 4 0 4 5 − 4 0 − 4 5 )

设矩阵<img src='https://img2.soutiyun.com/ask/2020-08-17/966509740284248.png' />,若向量a=(1, 1, k)^T是矩阵A^-1的对应于特征值λ的一个特征向量，求λ和k的值.

设<img src='https://img2.soutiyun.com/ask/2021-01-12/97930803924693.jpg' />，其中a＞0，n≥2。试问，当λ取何值时，实二次型x^TAx正定？

A．|A|&gt;0 B．负惯性指数为0 C．A的所有对角元a_ii&gt;0 D．A合同于单位阵I

设实二次型<img src='https://img2.soutiyun.com/ask/2021-01-06/978797001811459.jpg' /><img src='https://img2.soutiyun.com/ask/2021-01-06/978797017600362.jpg' />，证明：f(x₁，x₂，...，x_n)的秩等于矩阵。 <img src='https://img2.soutiyun.com/ask/2021-01-06/978797059757774.jpg' /> 的秩。

设A为n阶实对称矩阵，R(A)=n; 二次型 <img src='https://img2.soutiyun.com/ask/2020-12-17/977073420019754.png' /> (1)求二次型f的阵 (2) 二次型<img src='https://img2.soutiyun.com/ask/2020-12-17/977073445368204.png' />的规范形是否相同？说明理由.