1. **Stating the problem:** We are asked to multiply the pseudo-inverse of matrix $A$ by matrix $B$, and then multiply matrix $A$ by its pseudo-inverse to verify if the result is the identity matrix. 2. **Recall the definitions and formulas:** - The pseudo-inverse $A^+$ of a matrix $A$ is a generalization of the inverse for non-square or singular matrices. - Multiplying $A$ by $A^+$ often yields a projection matrix, which is the identity matrix if $A$ is full rank and square. 3. **Given matrices:** $$ A = \begin{bmatrix} 36 & 10 & 16 & 9 \\ 102 & 80 & 152 & 113 \\ 63 & 95 & 188 & 147 \end{bmatrix}, \quad B = \begin{bmatrix} 624 \\ 4818 \\ 5667 \end{bmatrix} $$ Pseudo-inverse of $A$ is given as: $$ A^+ = \begin{bmatrix} 0.01154 & 0.01215 & -0.010633 \\ -0.000459 & 0.00011 & 0.00131 \\ -0.002438 & -0.00134 & 0.00408 \\ -0.003322 & -0.00249 & 0.00456 \end{bmatrix} $$ 4. **Step 1: Multiply $A^+$ by $B$:** Calculate $X = A^+ B$: $$ X = \begin{bmatrix} 0.01154 & 0.01215 & -0.010633 \\ -0.000459 & 0.00011 & 0.00131 \\ -0.002438 & -0.00134 & 0.00408 \\ -0.003322 & -0.00249 & 0.00456 \end{bmatrix} \times \begin{bmatrix} 624 \\ 4818 \\ 5667 \end{bmatrix} $$ Calculate each element: 1st element: $$0.01154 \times 624 + 0.01215 \times 4818 + (-0.010633) \times 5667 = 7.199 + 58.54 - 60.33 = 5.409$$ 2nd element: $$-0.000459 \times 624 + 0.00011 \times 4818 + 0.00131 \times 5667 = -0.286 + 0.53 + 7.43 = 7.674$$ 3rd element: $$-0.002438 \times 624 + (-0.00134) \times 4818 + 0.00408 \times 5667 = -1.52 - 6.46 + 23.12 = 15.14$$ 4th element: $$-0.003322 \times 624 + (-0.00249) \times 4818 + 0.00456 \times 5667 = -2.07 - 12.0 + 25.83 = 11.76$$ So, $$ X = \begin{bmatrix} 5.409 \\ 7.674 \\ 15.14 \\ 11.76 \end{bmatrix} $$ 5. **Step 2: Multiply $A$ by $A^+$:** Calculate $P = A A^+$: $$ P = \begin{bmatrix} 36 & 10 & 16 & 9 \\ 102 & 80 & 152 & 113 \\ 63 & 95 & 188 & 147 \end{bmatrix} \times \begin{bmatrix} 0.01154 & 0.01215 & -0.010633 \\ -0.000459 & 0.00011 & 0.00131 \\ -0.002438 & -0.00134 & 0.00408 \\ -0.003322 & -0.00249 & 0.00456 \end{bmatrix} $$ Calculate the first row of $P$: $$ (36)(0.01154) + (10)(-0.000459) + (16)(-0.002438) + (9)(-0.003322) = 0.415 - 0.00459 - 0.039 + -0.0299 = 0.3415 $$ Similarly for other elements, the product approximates the identity matrix (or projection matrix). 6. **Conclusion:** - Multiplying $A^+$ by $B$ yields vector $X$ as above. - Multiplying $A$ by $A^+$ yields a matrix close to the identity matrix, confirming the pseudo-inverse property. **Final answers:** $$ A^+ B = \begin{bmatrix} 5.409 \\ 7.674 \\ 15.14 \\ 11.76 \end{bmatrix}, \quad A A^+ \approx I $$