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The dimension of the vector space of 10 x 10 real matrices with row sum, column sum, trace and sum of all entire as zero is 1.79 2. 78 3. 80 4. none.
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Deepak singh 1![best-answer]()
Consider any one row. since row sum is zero, so at least one component of that row can be written as the linear combination of other components. so that one element can be removed or can be replaced by zero. do this process for each row. Similarly, do this process for each column. you will find finally (10-1).(10-1) remaining elements. So the dimension is 9×9= 81. Now since trace is 0 , then one element from diagonal entries can be written with the help of other diagonal entry . So now dimension will become = 81-1= 80. Now , entire sum of all elements are 0, this implies one element of remaining matrix can be written with help of other remaining entries . So , now dimension is = 80-1= 79. thus answer is 79.