This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
a) To find the value of x, we use the formula for the determinant of a 2 × 2 matrix. Given the matrix A = 5 & -2 \\ -4 & x and its determinant is 7. Step 1: Write down the formula for the determinant of a 2 × 2 matrix. For a matrix a & b \\ c & d , the determinant is ad - bc. Step 2: Substitute the values from the given matrix into the determinant formula and set it equal to 7. (5)(x) - (-2)(-4) = 7 Step 3: Simplify and solve for x. 5x - 8 = 7 5x = 7 + 8 5x = 15 x = (15)/(5) x = 3 b) To find the inverse of the matrix, we first substitute the value of x found in part (a) into the matrix. Step 1: Write the matrix with the value of x. The matrix is now A = 5 & -2 \\ -4 & 3 . The determinant is given as 7. Step 2: Write down the formula for the inverse of a 2 × 2 matrix. For a matrix A = a & b \\ c & d , the inverse is A^-1 = (1)/((A)) d & -b \\ -c & a . Step 3: Substitute the values from the matrix and its determinant into the inverse formula. Here, a=5, b=-2, c=-4, d=3, and (A)=7. A^-1 = (1)/(7) 3 & -(-2) \\ -(-4) & 5 Step 4: Simplify the matrix. A^-1 = (1)/(7) 3 & 2 \\ 4 & 5 A^-1 = (3)/(7) & (2)/(7) \\ (4)/(7) & (5)/(7) Send me the next one 📸