Mathematics – Linear algebra I: matrix algebra
“What is the matrix?”
1. Find a 2 ⇥ 2 matrix X such that X² = I2. [2]
2. Verify that if X is the matrix you obtained in 1, and A = aI2 + bX for scalars a and b (not both 0), then A is invertible and there are scalars d and c such that A1 = dI2 + cX. [3]
3. Find 4 ⇥ 4 matrices U, V, and W such that U² = V² = W² = I4, UV = W, VU = W, VW = U, WV = U, WU = V, and UW = V. [2]
4. Verify that if U, V, and W are the matrices you obtained in 3, and B = aI4 + bU + cV+dW for scalars a, b, c, and d (not all 0), then B is invertible and there are scalars p, q, r, and s such that B¯¹ = pI4 + qU + rV + sW. [3]
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