矩阵单项选择题
1. 设A、B为同阶可逆矩阵, 则
(A) AB = BA (B) 存在可逆矩阵P, 使 ![](https://wkrtcs.bdimg.com/rtcs/image?w=76&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"76","h":"20","dataType":"png","c":"word/media/image1.png","imgOriW":76,"imgOriH":20})
(C) 存在可逆矩阵C, 使 ![](https://wkrtcs.bdimg.com/rtcs/image?w=75&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"75","h":"21","dataType":"png","c":"word/media/image2.png","imgOriW":75,"imgOriH":21})
(D) 存在可逆矩阵P和Q, 使
解. 因为A可逆, 存在可逆 ![](https://wkrtcs.bdimg.com/rtcs/image?w=136&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"136","h":"23","dataType":"png","c":"word/media/image4.png","imgOriW":136,"imgOriH":23})
.
因为B可逆, 存在可逆 ![](https://wkrtcs.bdimg.com/rtcs/image?w=136&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"136","h":"23","dataType":"png","c":"word/media/image5.png","imgOriW":136,"imgOriH":23})
.
所以 ![](https://wkrtcs.bdimg.com/rtcs/image?w=49&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"49","h":"23","dataType":"png","c":"word/media/image6.png","imgOriW":49,"imgOriH":23})
= . 于是
令 ![](https://wkrtcs.bdimg.com/rtcs/image?w=69&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"69","h":"24","dataType":"png","c":"word/media/image9.png","imgOriW":69,"imgOriH":24})
, . (D)是答案.
2. 设A、B都是n阶可逆矩阵, 则 ![](https://wkrtcs.bdimg.com/rtcs/image?w=100&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"100","h":"53","dataType":"png","c":"word/media/image11.png","imgOriW":100,"imgOriH":53})
等于
(A) ![](https://wkrtcs.bdimg.com/rtcs/image?w=105&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"105","h":"24","dataType":"png","c":"word/media/image12.png","imgOriW":105,"imgOriH":24})
(B) (C) (D)
解. ![](https://wkrtcs.bdimg.com/rtcs/image?w=217&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"217","h":"53","dataType":"png","c":"word/media/image16.png","imgOriW":217,"imgOriH":53})
. (A)是答案.
3. 设A、B都是n阶方阵, 下面结论正确的是
(A) 若A、B均可逆, 则A + B可逆. (B) 若A、B均可逆, 则AB可逆.
(C) 若A + B可逆, 则A－B可逆. (D) 若A + B可逆, 则A, B均可逆.
解. 若A、B均可逆, 则 ![](https://wkrtcs.bdimg.com/rtcs/image?w=108&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"108","h":"24","dataType":"png","c":"word/media/image17.png","imgOriW":108,"imgOriH":24})
. (B)是答案.
4. 设n维向量 ![](https://wkrtcs.bdimg.com/rtcs/image?w=119&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"119","h":"41","dataType":"png","c":"word/media/image18.png","imgOriW":119,"imgOriH":41})
, 矩阵 , 其中E为n阶单位矩阵, 则AB =
(A) 0 (B) －E (C) E (D) ![](https://wkrtcs.bdimg.com/rtcs/image?w=61&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"61","h":"21","dataType":"png","c":"word/media/image21.png","imgOriW":61,"imgOriH":21})
解. AB = ![](https://wkrtcs.bdimg.com/rtcs/image?w=72&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"72","h":"24","dataType":"png","c":"word/media/image22.png","imgOriW":72,"imgOriH":24})
= + 2 －2
= E. ![](https://wkrtcs.bdimg.com/rtcs/image?w=73&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"73","h":"41","dataType":"png","c":"word/media/image26.png","imgOriW":73,"imgOriH":41})
(C)是答案.
5. 设 ![](https://wkrtcs.bdimg.com/rtcs/image?w=135&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"135","h":"75","dataType":"png","c":"word/media/image27.png","imgOriW":135,"imgOriH":75})
, , , 设有P2P1A = B, 则P2 =
(A) ![](https://wkrtcs.bdimg.com/rtcs/image?w=71&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"71","h":"75","dataType":"png","c":"word/media/image30.png","imgOriW":71,"imgOriH":75})
(B) (C) (D)
解. P1A表示互换A的第一、二行. B表示A先互换第一、二行, 然后将互换后的矩阵的第一行乘以(－1)加到第三行. 所以P2 = ![](https://wkrtcs.bdimg.com/rtcs/image?w=81&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"81","h":"75","dataType":"png","c":"word/media/image31.png","imgOriW":81,"imgOriH":75})
.(B)是答案.
6. 设A为n阶可逆矩阵, 则(－A)*等于
(A) －A* (B) A* (C) (－1)nA* (D) (－1)n－1A*
解. (－A)* = ![](https://wkrtcs.bdimg.com/rtcs/image?w=296&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"296","h":"44","dataType":"png","c":"word/media/image34.png","imgOriW":296,"imgOriH":44})
. (D)是答案.
7. 设n阶矩阵A非奇异(n 2), A*是A的伴随矩阵, 则
(A) ![](https://wkrtcs.bdimg.com/rtcs/image?w=107&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"107","h":"24","dataType":"png","c":"word/media/image35.png","imgOriW":107,"imgOriH":24})
(B)
(C) ![](https://wkrtcs.bdimg.com/rtcs/image?w=108&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"108","h":"24","dataType":"png","c":"word/media/image37.png","imgOriW":108,"imgOriH":24})
(D)
解. ![](https://wkrtcs.bdimg.com/rtcs/image?w=83&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"83","h":"24","dataType":"png","c":"word/media/image39.png","imgOriW":83,"imgOriH":24})
![](https://wkrtcs.bdimg.com/rtcs/image?w=463&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"463","h":"24","dataType":"png","c":"word/media/image40.png","imgOriW":463,"imgOriH":24})
(C)是答案.
8. 设A为m×n矩阵, C是n阶可逆矩阵, 矩阵A的秩为r1, 矩阵B = AC的秩为r, 则
(A) r > r1 (B) r < r1 (C) r = r1 (D) r与r1的关系依C而定
解. ![](https://wkrtcs.bdimg.com/rtcs/image?w=164&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"164","h":"24","dataType":"png","c":"word/media/image41.png","imgOriW":164,"imgOriH":24})
, 所以
![](https://wkrtcs.bdimg.com/rtcs/image?w=209&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"209","h":"23","dataType":"png","c":"word/media/image42.png","imgOriW":209,"imgOriH":23})
又因为 ![](https://wkrtcs.bdimg.com/rtcs/image?w=65&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"65","h":"21","dataType":"png","c":"word/media/image43.png","imgOriW":65,"imgOriH":21})
, 于是
![](https://wkrtcs.bdimg.com/rtcs/image?w=233&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"233","h":"24","dataType":"png","c":"word/media/image44.png","imgOriW":233,"imgOriH":24})
所以 ![](https://wkrtcs.bdimg.com/rtcs/image?w=39&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"39","h":"23","dataType":"png","c":"word/media/image45.png","imgOriW":39,"imgOriH":23})
. (C)是答案.
9. 设A、B都是n阶非零矩阵, 且AB = 0, 则A和B的秩
(A) 必有一个等于零 (B) 都小于n (C) 一个小于n, 一个等于n (D) 都等于n
解. 若 ![](https://wkrtcs.bdimg.com/rtcs/image?w=283&md5sum=abd6431abdcfa929d58061848e3c6c0d&sign=6f91d2afd5&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=475&ipr={"t":"img","w":"283","h":"24","dataType":"png","c":"word/media/image46.png","imgOriW":283,"imgOriH":24})
, 矛盾. 所以 . 同理 . (B)是答案.
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