Textbook Question
In Exercises 13 - 18, use the fact that ifa b d - bA = then A^(-1) = 1/(ad-bc) to find the inverse of c d - c aeach matrix, if possible. Check that AA^(-1) = I_2 and A^(-1)A = I_2. 10 - 2A = - 5 1
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7. Systems of Equations & Matrices
Determinants and Cramer's Rule
7:49 minutesProblem 40
Textbook Question
In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I
Verified step by step guidance1
Step 1: Recall the definition of the inverse of a matrix. A matrix A has an inverse A^(-1) if and only if A is a square matrix (same number of rows and columns) and its determinant is non-zero. The inverse satisfies the property that AA^(-1) = I and A^(-1)A = I, where I is the identity matrix.
Step 2: Compute the determinant of the matrix A. For a 2x2 matrix A = [[a, b], [c, d]], the determinant is given by det(A) = ad - bc. If the determinant is zero, the matrix does not have an inverse.
Step 3: If the determinant is non-zero, calculate the inverse of the matrix. For a 2x2 matrix A = [[a, b], [c, d]], the inverse is given by A^(-1) = (1/det(A)) * [[d, -b], [-c, a]]. Replace the elements of the matrix and the determinant into this formula to find A^(-1).
Step 4: Verify that AA^(-1) = I by performing matrix multiplication. Multiply the original matrix A by its inverse A^(-1) and confirm that the result is the identity matrix I.
Step 5: Similarly, verify that A^(-1)A = I by performing matrix multiplication in the reverse order. Multiply the inverse A^(-1) by the original matrix A and confirm that the result is the identity matrix I.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse of a Matrix
The inverse of a matrix A, denoted A^(-1), is a matrix that, when multiplied by A, yields the identity matrix I. This means that A * A^(-1) = I and A^(-1) * A = I. Not all matrices have inverses; a matrix must be square and have a non-zero determinant to possess an inverse.
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Identity Matrix
The identity matrix, denoted I, is a special square matrix that has ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication, meaning that for any matrix A, multiplying by I leaves A unchanged (A * I = A). The identity matrix is crucial for verifying the correctness of matrix inverses.
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Matrix Multiplication
Matrix multiplication involves combining two matrices to produce a third matrix. The number of columns in the first matrix must equal the number of rows in the second. The resulting matrix's elements are calculated by taking the dot product of the rows of the first matrix with the columns of the second, which is essential for checking the properties of inverses.
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