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Which of the following are true? (1) The sum of two invertible matrices is invertible (2) The...

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Basic Math

Which of the following are true?

(1) The sum of two invertible matrices is invertible

(2) The determinant of the sum is the sum of thedeterminants

(3) The determinant of the inverse is the reciprocal of thedeterminant

(4) An nxn matrix is invertible if and only if its determinantis zero

(5) The product of two invertible matrices is invertible (solong as the product is defined)

(6) If A is a diagonal 3x3 matrix [a,0,0;0,b,0;0,0,c] then itsdeterminant is the product of its diagonal entries

(7) The union of two subspaces is a subspace

(8) The intersection of two subspaces is a subspace

(9) The set of solutions [x,y,z,u,v] to the equations 3x + 4y +5z + u + 3v = 12 and 4x + y + z + u - v = 13 form a vectorspace

(10) If V is a vector space of polynomials and p and q are in Vthen the product pq is also in V

(11) If V is a subspace of R^3 and w is a vector in v thenwhenever the dot product w dot u = 0, u must also be in thesubspace V

(matrices are written as e.g. [a,b,c; d,e,f; g,h,i; j,k,l] tomean the matrix whose first row is [a,b,c], whose second row is[d,e,f] whose third row is [g,h,i] and whose fourth row is[j,k,l].)

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