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

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 the determinants

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

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

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

(6) If A is a diagonal 3x3 matrix [a,0,0;0,b,0;0,0,c] then its determinant 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 vector space

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

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

(matrices are written as e.g. [a,b,c; d,e,f; g,h,i; j,k,l] to mean 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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milcah answered 1 year ago

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