Matrix by Scalar Calculator

To multiply a matrix by a scalar, simply multiply every single element in the matrix by that scalar value. For example, if your scalar is 3 and your matrix contains the element 4, the resulting element is 3 × 4 = 12. The dimensions of the matrix stay exactly the same.

Scalar multiplication follows several key properties: it is distributive over matrix addition — k(A + B) = kA + kB; it is associative — (xy)A = x(yA); multiplying by 1 leaves the matrix unchanged — 1·A = A; and multiplying by 0 produces a zero matrix. These properties make scalar multiplication a fundamental operation in linear algebra.

Dividing a matrix by a number n is the same as multiplying it by 1/n. Simply enter your divisor as a fraction (e.g., 0.5 to divide by 2) in the scalar field. Every element of the matrix will be divided by that number.

If you multiply an n×n matrix A by a scalar k, the determinant of the result is kⁿ · det(A). For example, multiplying a 3×3 matrix by scalar k gives a determinant of k³ · det(A). This is because each row of the determinant picks up a factor of k.

When you multiply a matrix A by scalar k, the eigenvalues of the resulting matrix kA are simply k times each eigenvalue of A. The eigenvectors, however, remain unchanged. So if λ is an eigenvalue of A, then kλ is an eigenvalue of kA.

Any matrix multiplied by the scalar 0 results in a zero matrix — a matrix of the same dimensions where every element equals 0. This is consistent with the property that 0 × anything = 0.

Multiplying the identity matrix I by a scalar k produces a scalar matrix — a diagonal matrix where each diagonal element equals k and all off-diagonal elements remain 0. This matrix acts like k when multiplied with other conformable matrices.

No, scalar multiplication is commutative with respect to the scalar itself. k·A = A·k, meaning the scalar can be written on either side of the matrix. However, this only applies to scalar multiplication; regular matrix-by-matrix multiplication is generally not commutative.